WC0MM£m 


mRifTHMm-ic, 


..>^oTATio?f.  [Sect.  I. 

6.  One  thousand,  and  forty-two. 

7.  ThMy  thousand,  nine  hundred  and  seven. 
8    Forty-six  thousand,  and  four  hundred. 

'  9'.  Ninety-two  thousand,  one  hundred  and  eight. 
10"  Sixty-eight  thousand,  and  seventy.  .     ,       i     1 

11!  One  liundrcd  and  twenty-four  thousand,  six  hundred 

and  thirty.  ^    -  .^ 

12.  Two  hundred  thousand,  one  hundred  and  sixt}^. 

13.  Four  hundred  and  five  thousand,  and  forty -five. 
14    Three  hundred  and  forty  thousand. 

15.  Nine  hundred  thousand,  seven  liundred  and  twenty. 

16    One  railhon,  and  seven  hundred  thousand.  ' 

17.  Tdiirty-six  railhons,^ twenty  thousand,  one  liundred  and 

18    One  hundred  millions,  and  forty-five. 

19.  Mercury  is  thirty-seven  millions  of  miles  from  the  sun. 

20.  Venus,  sixty-nine  millions. 

21.  Tlie  Earth,  ninety -five  millions. 

22.  Mars,  one  liwi.vi.oa  otiJ  foT-fy-five  millions. 

23.  Jupiter,  four  hundred  and  ninety-four  millions. 

24.  Saturn,  nine  hundred  and  seven  millions. 

25.  Ilerschel,  one  hiUion,  eiglit  hundred  and  ten  millions. 

26.  Seven  billions,  nine  hundred  millions,  and  forty  thousand. 

27.  Sixty  billions,  seven  millions,  and  four  hundred. 

28.  One  hundred  and  thirteen  billions,  six  hundred  and  fifty 
thousand. 

29.  Four  hundred  and  six  billions,  eight^^^^aillions,  and  seven 
hundred. 

30.  Twenty-five  trillions,  and  te^-.  thousan^l.  , 

31.  Two  Hundred  and  six  billions,  five  hundred  and  sixty 
thousand,  and  forty-five. 

32.  Six  hundred  millions,  seventeen  thousand,  three  hun- 
dred  and  eight. 

83.  Ninety-seven  trillions,  sixteen   millions,  seventy  th®u- 
eand,  and  thirty. 

34.  Eight  hundred  and  forty  billions,  fifty  mil^ons,  three 

hundred  and  one  thousand.  \     -  ,_v^,^ty— 

35.  Tiiree  hundred  and  sixfAT-fivp,  nnn/lrininr!'»  ft# 


(D-f 


University  of  California  •  Berkeley 


The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


MERCHANTS  &  MECHANICS' 
COM]\IERCIAL 

AEITHMETIC ; 

OR, 

INSTANTANEOUS  METHOD  OF 
COMPUTIL^TG  NUMBERS. 


By   JOHN    E.    WADE. 


Ifew  York  : 

AUSSBLL    BROTHERS,     PUBLISHERS, 

17,  I'J,  21,  ^.SRoa/.S.iiKu:-. 

1373. 


Entered  according  to  Act  of  Congress,  in  the  year  1870, 

By  JOHN  E.  WADE, 

in  the  Office  of  the  Librarian  of  Congress,  at  Washington,  D.  C. 


PREFACE 


Hathe3£atical  LAWS  are  the  acknowledged  basis 
of  all  science.  Ever  since  the  streets  of  Athens 
resounded  with  that  historical  cry  of  "Eureka," 
emanating  from  one  of  antiquity's  greatest  mathe- 
maticians, the  science  has  been  steadily  progressing. 

It  is  not  our  purpose  in  this  small  work  to  intro- 
duce any  of  the  higher  branches  of  mathematics, 
viz..  Algebra,  Conic  Sections,  Calculus,  etc.  Our 
object  is  merely  to  present  to  the  public  a  sys- 
tem of  calculation  that  is  practical  to  every  business 
man.  It  consists  of  the  addition  of  numbers  on  a 
principle  entirely  different  from  the  one  ordinarily 
used-  In  the  practical  application  of  this  new  prin- 
ciple of  Addition  scarcely  any  mental  labor  is  re- 
quired, compared  with  the  principle  of  Addition  set 
forth  in  standard  works.  The  superiority  we  claim 
for  this  principle  above  all  others  is  this — that  it 
requires  no  great  mental  exertion,  affording  the 
<jreatest  facilities  to  the  calculator  in  the  addition 
pf  numbers,  enabling  him  to  add  a  whole  day  with- 
Dut  any  mental  fatigue  ;  whereas,  by  the  ordinary 
way,  it  is  very  laborious  and  fatiguing. 

Our  system  of  calculation  also  embraces  a  concise, 


PREFACE. 

rapid,  and,  at  the  same  time,  practical  method  of 
Multiplication,  by  which  one  is  enabled  to  arrive  at 
the  product  of  any  number  of  figures,  multiplied  by 
any%um]ber,  immediately,  without  the  use  of  partial 
products. 

This  small  work  also  embraces  the  shortest  and 
most  concise  method  for  the  computation  of  Interest 
ever  introduced  to  the  public.  Our  system  for  com- 
puting interest  is  entirely  different  from  any  rule 
ever  introduced  for  the  computation  of  either  Sim- 
ple or  Compound  Interest.  A  student,  having  gone 
no  further  than  Long  Division  in  Arithmetic,  can, 
by  our  rule,  calculate  Simple  or  Compound  Interest 
at  any  given  rate  per  cent.,  for  any  given  time,  in 
one  tenth  of  the  time  that  the  best  calculators  will 
compute  it  by  the  rules  laid  down  in  other  books. 
By  using  our  rules  you  can  entirely  avoid  the  use 
of  fractions,  and  save  the  calculation  of  75  to  100 
figures  where  years,  months  and  day^j  are  given  on 
a  note. 


INTRODUCTION. 


Quantity  is  that  which  can  be  increased  or  dimin- 
ished by  augments  or  abatements  of  homogeneous 
parts.  Quantities  are  of  two  essential  kinds,  Geo- 
metrical and  Physical. 

1.  Geometrical  quantities  are  those  which  occupy 
space — as  lineSj  surfaces,  solids,  liquids,  gases,  etc. 

2.  Physical  quantities  are  those  which  exist  in  the 
time,  but  occupy  no  space  ;  they  are  known  by  their 
character  and  action  upon  geometrical  quantities,  as 
attraction,  light,  heat,  electricity  and  magnetism,  colors, 
force,  power,  etc. 

To  obtain  the  magnitude  of  a  quantity  we  com- 
pare it  with  a  part  of  the  same  ;  this  part  is  im- 
printed in  our  mind  as  a  unit,  by  which  the  whole 
is  measured  and  conceived.  No  quantity  can  be 
measured  by  a  quantity  of  another  kind,  but  any 
quantity  can  be  compared  with  any  other  quantity, 
and  by  such  comparison  arises  what  we  call  calcula- 
tion or  Mathematics. 

MATHEMATICS. 

Mathematics  is  a  science  by  which  the  compara- 
tive value  of  quantities  is  investigated  ;  it  is  divided 


INTRODUCTION. 

1.  Arithmetic,  that  branch  of  Mathematics  which 
treats  of  the  nature  and  property  of  numbers.  It  is 
subdivided  into  Addi'tion,  Subtraction,  Multiplication, 
DwrnoHf    Involution,  Evolution  and  Logarithms. 

2.  Algebra,  that  branch  of  Mathematics  which 
employs  letters  to  represent  quantities,  and  by  that 
means  performs  solutions  without  knowing  or  notic- 
ing the  value  of  the  quantities.  The  subdivisions  of 
Algebra  are  the  same  as  in  Arithmetic. 

3.  Geometry,  that  branch  of  Mathematics  which 
investigates  the  relative  property  of  quantities  that 
occupies  space  ;  its  subdivisions  are  Longemetry, 
Planemetry,  Stereometry,  Trigonometry  and  Conic 
Sections. 

4.  Differential-calculus,  that  branch  of  Mathema- 
tics which  ascertains  the  mean  effect  produced  by 
group  of  continued  variable  causes. 

.5.  Integral-calculus,  the  contrary  of  Differential, 
or  that  branch  of  Mathematics  which  investigates 
the  nature  of  a  continued  variable  cause  that  has 
produced  a  known  effect. 


NOTATION 

is  the  rule  for  writing  numbers.  The  Arabic  charac- 
ters or  figures  are  1,  2,  3,  4,  5,  6,  T,  8,  9,  0.  These  fig- 
ures have  two  values,  viz.,  the  simple  value — its  value 
when  taken  alone — and  the  local  value — its  value 
when  used  with  another  figure  or  figures.  Counting 
from  the  right  hand  the  first  figure  expresses  units, 
the  second  tens,  the  third  hundreds,  and  every  remo- 
val of  a  figure  one  place  towards  the  left  increases  its 
local  value  ten  times. 

NUMERATION 

is  the  rule  for  reading  numbers.  According  to  the 
French  method — the  one  generally  in  use — every 
three  figures  constitute  a  period.  Commencing  upon 
the  right,  the  periods  are  successively  named  units, 
thousands,  millions,  billions,  trillions,  quadrillions, 
quintillions,  sextillions,  septillions,  octillions,  nonil- 
lions,  decillions,  undecillions,  duodecillions,  tredecil- 
lions,  quartodecillions,  quindecillions,  sexdecillions, 
septdecillions,  octodecillions,  nondecillions,  virgitil- 
lions.  By  committing  the  names  of  these  periods  to 
memory  you  will  be  enabled  to  enumerate  sixty-six 
figures — a  number  far  greater  than  will  ever  be  re- 
required  in  practice.  Beginning  at  the  left  hand, 
read  each  period  separately,  giving  the  name  to  each 
period  except  the  last,  or  period  of  units. 


8  the  merchants  and  mechanics* 

Example.      3T,  546,  829,  106,  391,  463,  512,  460,  831 


Read — thirty-seven  septillions,  five  hundred  and 
forty-six  sextillions,  eight  hundred  and  twenty-nino 
quintillions,  etc. 

SIGNS  USED  IN  MATHEMATICS. 

The  sign=  is  called  the  sign  of  equality.  When 
placed  between  two  numbers  it  signifies  that  they 
are  equal  to  each  other. 

The  sign  -{-  is  called  pZws,  which  signifies  more. 
When  placed  between  two  numbers  it  denotes  that 
they  are  to  be  added  together. 

The  sign  —  is  called  minus,  which  signifies  less. 
When  placed  between  two  numbers  it  denotes  that 
the  one  after  it  is  to  be  taken  from  the  one  before  it. 

The  sign  X>  when  placed  between  two  numbers, 
denotes  that  they  are  to  be  multiplied  together. 

The  sign  -^-,  placed  between  two  numbers,  de- 
notes that  the  number  before  it  is  to  be  divided  by 
the  number  after  it.  Division  is  also  expressed  by 
writing  the  dividend  above  a  short  horizontal  line 
and  the  divisor  below  it,  thus  :  ^^^ 

The  sign  $,  contraction  of  U.  S.,  denotes  dollars, 
or  United  States  currency. 

The  sign  ^  is  called  the  radical  sign,  which, 
placed  before  a  number,  denotes  that  its  square  root 
is  to  be  extracted. 

ADDITION 

is  uniting  two  or  more  numbers  to  find  their  sum  or 
amount. 
Addition  is  the  basis  of  all  numerical  calculations. 


COMMERCIAL    ARITHMETIC.  9 

and  is  used  in  all  departments  of  business.  To  be 
able  to  add  with  ease,  precision  and  rapidity,  is  a 
great  desideratum  by  every  accountant.  To  aid  the 
business  man  in  acquirincr  facility  and  accuracy  in 
adding  numbers  the  following  methods  are  pre- 
sented as  among  the  best.  But  it  may  as  well  be 
understood  that  no  method  can  be  presented  that 
will  entirely  do  away  with  all  labor  of  the  brain — for 
to  add  accurately  and  rapidly  can  only  be  acquired 
by  close  application  and  careful,  constant  practice. 
In  writing  down  the  numbers  for  addition,  those  of 
the  same  order  should  be  written  exactly  under  each 
other.  Too  much  care  cannot  be  used  in  this  re- 
spect. Many  serious  blunders  are  often  made  from 
no  other  cause  than  the  bad  habit  of  writing  num- 
bers carelessly,  getting  units  under  tens  or  tens 
under  units,  etc.  Perhaps,  too,  this  would  be  as 
proper  a  time  as  any  to  urge  the  importance  of  mak- 
ing jMin  firjures.  If  letters  are  badly  made  you  can 
judge  from  such  as  are  known,  but  if  one  figure 
be  illegible  it  cannot  be  inferred  from  the  others. 
Making  figures  is  a  habit  ;  therefore,  a  little  care  at 
first  in  making  them  jylain,  may  save  much  anxious 
toil  and  serious  trouble  afterward.  After  having 
written  the  numbers,  those  of  the  same  order  exactly 
under  each  other,  begin  with  the  right  hand,  or 
unit  column,  and  proceed  to  add  together  all  the 
figures  in  that  column,  and  write  down  the  entire 
amount  underneath.  Then  carry  or  add  all  except 
the  unit  figure  of  that  amount  to  the  next  column, 
adding  yp  that  column  as  before,  and  write  the  en- 
tire amount  underneath  the  former  one.  Proceed  in 
this  manner  with  each  column  until  all  are  added, 
setting  down  the  entire  amount  of  each  column,  one 
under  the  other.  The  last  amount,  with  the  unit 
figure  of  each  amount  annexed,  will  be  the  correct 
amount  of  the  entire  Bum. 


10  THE    MERCHANTS    AND    MECHANICS' 

Example.        Process. — Commencing  at  4  in  the  unit 
3462         column,  say  4,  9,  15,  18,  19,  26,  29,  35,  39, 
895         44,  46,   setting  down  the  entire  amount. 
*7604         Then  add  the  four  tens  of  this  amount  to 
•9346         to  the  cohimns  of  tens  thus  :  4,  t,   9,  13, 
8753         15,  21,  23,  28,  32,  41,  41,  setting  down  the 
4927         amount.    Then  carry  the  4  of  this  amount 
2861         to  the  next  column,  and  say  4,  13,  20,  28, 
920         37,  45,  54,  61,  64,  70,  78,  82,  setting  down 
803         the  amount  as  before.     Then  carrying  the 
6746         8  to  the  next  column,  say  8,  15,  21,  23,  27, 
25         35,  44,  51,  54.     Then   this  last   amount, 
7934         with  the  unit  figure  of  each  amount  an- 
nexed in  the  order  as  they 
occur,  reading  upward,  will 
be  the  entire  amount.     The 
partial  amounts  can  be  set 
down  upon  a  separate  piece 
of  paper,  if  desired,  and  only 
54276  entire  amount.       the  entire  amount  written 
under  the  sum.     The  prac- 
tice, however,  of  setting  down  the  amount  of  each 
column  will  be  found  highly  convenient — as,  knowing 
what  number  you  carried,  you  will  be  able  at  any 
time  to  add  any  column  without  being  obliged  to 
add  all  that  precede  it. 

THE  EASY  WAY  TO  ADD. 

Example  2. — Explanation. 

Commence  at  9  to  add,  and  add  as  near  20  as 
possible,  thus:  9-|-2+4+3~18 ;  place  the  8  to 
the  right  of  the  3,  as  in  example  ;  commence  at  6  to 
add  6-f-4-f8=:18  ;  place  the  8  to  the  right  of  the  8, 
as  in  example  ;  commence  at  6  to  add  6-|-4-[-7  =  17  ; 
place  the   7   to  the  right  of  the  7,  as  in  example  ; 


46 

am' 

't  of  units. 

47 

<< 

"   tens. 

82 

tt 

"   hund. 

54 

it 

"   thou. 

COMMERCIAL    ARITHMETIC.  11 

commenee  .ft  4  to  add  4+9+3=16  ;  place  the  V 

6  to  the  right  of  the  3,  as  in  example  ;  commence  4 

at  6  to  add  6+4+7+1 1  ;  place  the  1  to  the  6 

right  of  the  7,  as  in  example.     Now,  having  ar-  3® 

rived  at  the  top  of  the  column,  we  add  the  figures  9 

in  the  new  column   thus  :  7+6+7+8+8=36  ;  4 

place  the  right  hand  figure  of  36,  which  is  a  6,  7^ 

under  the  original  column,  as  in  example,  and  4 

add  the  left  hand  figure,  which  is  a  3,  to  the  6 

number  of  figures  in  the  new  column  ;  there  are  8' 

5  figures  in  the  new  column,  therefore  3+5=8  ;  4 

prefix  the  8  with  the  6  under  the  original  column,  6 

as  in  example  ;  this  makes  86,  which  is  the  sum  3* 

of  the  column.  4 

Remark. — If,  upon  arriving  at  the  top  of  the  2 

column,  there  shoukl  be  one,  two  or  three  figures  9 

whose  sum  will  not  equal  10,  add  them  on  to  the  — 

sum  of  the  figures  of  the  new  column,  never  pla-  86 
cing  an  extra  figure  to  the  new  column  unless 
it  be  an  excess  of  units  over  10. 

Example. — Explanation. 
2+6+7=15,  drop  10,  place  the  5  to  the  right  4 
of  the  7  ;  6+5+4=15,  drop  10,  place  the  5  to  7* 
the  right  of  the  4,  as  in  example  ;  8+3+7=:18,.  3 
drop  10,  place  the  8  to  the  right  of  the  7,  as  in  8 
example  ;  now  we  have  an  extra  figure,  which  is  4" 
4  ;  add  this  four  to  the  top  figure  of  the  new  col-  5 
umn,  and  this  sum  on  the  balance  of  the  figures  6 
in  the  new  column,  thus:  4+8+5+5=22;  7' 
l)hice  the  right  hand  figure  of  22  under  the  ori-  6 
pinal  cohimn,  as  in  example,  and  add  the  left  2 
liand  figure  of  22  to  the  number  of  figures  in  the  — 
new  cohimn,  which  are  three,  thus  :  2+3=5  ;  62 
prefix  this  five  to  tlie  figured,  under  the  original  col^ 
umn  :  this  makes  52,  which  is  the  sum  of  the  column. 


12  THE    MERCHANTS    AND    MECHANICS' 


ANOTHER  METHOD. 

6  4  Process. — Commence   at  4  to  add,  and 

2'8^  add  as  near  20  as  possible,  thus:  4-|-2-j- 

3  2  3+7=16;  place  the  6  to  the  right  hand  of 

6  6  the  1,  as  in  example.     Commence  at  9  to 

3  1"  add,  9+1=16;  place  the  6  to  the  right  of 

4  9  the  7.     Commence  at  6  to  add,  6+2+=8 
W  16;  place  the  6  to  the  right  of  the  8,  as  in 

2  3         example.     Now,  having  arrived  at  the  top 
6  2         of  the  column,  or  so  near  that  the  remain- 

3  4         ing  figures  will  not  equal  10,  we  add  the 

figures  in  the  new  column,  thus:  6+6+6= 

402         18+4  (the  remaining  figure)=22.     Set  the 

unit  figure  under  the  column,  and  add  the 
left  hand  figure,  which  is  2,  to  the  number  of  figures 
in  the  new  column.  The  number  of  figures  in  the 
new  column  are  3,  therefore  2+3=5.  Carry  the  5 
to  the  3  in  the  next  column,  and  say,  5+3+6+2= 
1=17;  place  the  7  to  the  right  of  the  1,  as  in  exam- 
ple. Commence  at  4  to  add,  4+3+5+3+2=17  ; 
place  the  7  to  the  right  of  the  2.  Now,  having 
arrived  at  the  top  of  the  column,  or  so  near  that 
the  remaining  figures  will  not  equal  10,  we  add 
the  .figures  in  the  new  column,  7+7=14+6  (the 
remaining  figure)^20.  Set  the  right  hand  figure, 
which  is  0,  under  the  column,  and  add  the  left 
hand  figure,  2,  to  the  number  of  figures  in  the  new 
column,  which  are  2;  therefore  2+2=4,  wiiich  prefix 
to  the  amounts  already  set  down,  and  you  have  402, 
the  entire  amount. 

This  method  is  useful  only  in  adding  very  long 
columns  of  figures,  say  a  long  ledger  column,  where 
the  footings  of  each  column  would  be  two  or  three 
hundred,  in  which  case  it  is  more  easy  than  any 
other  mode,  as  the  mind  is  relieved  at  intervals,  and 


COMMERCIAL    ARITHMETIC.  13 

the  mental  labor  of  retaining  the  whole  amount  as 
you  add  is  avoided,  which  is  very  important  to  any 
person  whose  mind  is  constantly  employed  in  various 
commercial  calculations.  The  small  figures  in  the 
new  column  can  be  made  lightly  in  pencil,  so  as  to 
be  easily  erased,  and  the  ledger  will  only  show  the 
total  amounts. 

But,  whatever  method  may  be  used,  never  for  once 
allow  yourself  to  add  in  this  way  :  4  and  2  are  6, 
and  3  are  9,  and  t  are  16,  and  9  are  25,  and  7  are  32, 
etc.  It  is  just  as  easy  to  name  the  results  of  two 
figures  at  once,  and  four  times  as  rapid,  thus  :  4,  6, 
9^  16,  25,  32,  38,  40,  48,  52  ;  or,  better  still,  to  form 
the  habit  of  grouping  them,  thus  :  6,  16,  25,  32,  62. 
We  see  at  a  glance  that  4  and  2  are  6,  that  3  and  7 
are  10,  making  16,  and  9  are  25,  and  1  are  32  ;  that 
6  and  4  are  10,  and  that  2  and  8  are  10,  making  20, 
which  added  to  32  are  52.  By  practicing  this  method 
a  short  time  you  Avill  soon  acquire  a  proficiency  that 
will  astonish  the  uninitiated. 

The  method  of  proof  most  commonly  employed  is 
to  reverse  the  process  of  addition,  or  commence  at 
the  top  and  add  downward.  If  the  results  agree 
the  work  is  supposed  to  be  right. 

MULTIPLICATION. 
Multiplication,  in  its  most  general  sense,  is  a  series 
of  additions  of  the  same  number;  therefore,  in  Mul- 
tiplication, a  number  is  repeated  a  certain  number 
of  times,  and  the  result  thus  obtained  is  called  the 
product.  When  the  multiplicand  and  the  multiplier 
are  each  composed  of  only  two  figures,  to  ascertain 
the  product  we  have  the  following 

EuLE, — S(H  down  the  smaller  factor  under  the  larger, 
unit.-i  under  units,  tens  under  tens.  Begin  vnth  the 
un:t  figure  of  the  multiplier,  multiply  by  it  first  the 


14  THE   MERCHANTS   AND   MECHANICS* 

units  of  the  multiplicand,  setting  the  units  of  the  pro- 
duct,  and  resermng  the  tens  to  he  added  to  the  next 
product ;  noio  multiply  the  tens  of  the  multiplicand  by 
the  unit  figure  of  the  multiplier,  and  the  units  of  the 
multiplicand  by  tens  figure  of  the  multiplier ;  add 
these  two  products  together,  setting  down  the  units  of 
their  sum,  and  resermng  the  tens  to  he  added  to  the 
next  product ;  now  multiply  the  tens  of  the  multipli- 
cand by  the  tens  figure  of  the  multiplier,  and  set  down 
the  whole  amount.     This  will  be  the  complete  product. 

Remark. — Always  add  in  the  tens  that  are  reserved 
as  soon  as  you  form  the  first  product. 

Example  1. — Explanation. 
1.  Multiply  the  units  of  the  multiplicand  by       24 
the  unit  figure  of  the  multiplier,  thus  :  IX"^  i^       ^1 

4  ;  set  the  4  down  as  in  example.     2.  Multiply     

the  tens  in  the  multiplicand  by  the  unit  figure  144 
in  the  multiplier,  and  the  units  in  the  multipli- 
cand by  the  tens  figure  in  the  multiplier,  thus  :  IX 
2  is  2,  3X4  are  12;  add  these  two  products  together, 
2-}-12  are  14  ;  set  the  4  down  as  in  example,  and  re- 
serve the  1  to  be  added  to  the  next  product.  3.  Mul- 
tiply the  tens  in  the  multiplicand  by  the  tens  figure 
in  the  multiplier,  and  add  in  the  tens  that  were  re- 
served, thus  :  3X2  are  6,  and  6-|-l='T  ;  now  set 
down  the  whole  amount,  which  is  *I. 

Example  2. — Explanation. 

1.  Multiply  units  by  units,  thus  :  4X3  are         53 
12  ;  set  down  the  2  and  reserve  the  1  to  carry.         84 

2.  Multiply  tens  by  units   and  units  by  tens, 

and  add  in  the  1  to  carry  on  the  first  product,     4452 
then  add  these  two  products  together,  thus  : 
4X5  are  20+1  are  21,  and  8X3  are  24,  and  21+24 
are  45  ;  set  down  the  5  and  reserve  the  4  to  carry  to 


COMMERCIAL    ARITHMETIC.  15 

the  next  product,  3.  Multiply  tens  bv  tens,  and 
add  in  what  was  reserved  to  carry,  thus  :  8X5  are 
40-f-4  are  44  ;  now  set  down  the  whole  amount, 
which  is  44. 

Example  3. — Explanation. 

5X3  are  15,  set  down  the  5  and  carry  the  43 
1  to  the  next  product;  oX^  are  20-|-l  are  21;         25 

2X3  are  6,  21+6  are  27,  set  down  the  1  and     

carry  the  2  ;  2X4  are  8+2  are  10  ;  now  set  1075 
down  the  whole  amount. 

When  the  multiplicand  is  composed  of  three  fig- 
ures, and  there  are  only  two  figures  in  the  multiplier, 
we  obtain  the  product  by  the  following 

Rule. — Set  dovm,  the  smaller  factor  under  the  larger j 
units  under  units,  tens  under  tens  ;  noio  multiply  the 
first  upper  figure  by  the  unit  figure  of  the  multiplier, 
setting  down  the  units  of  the  product,  and  reserving 
the  tens  to  he  added  to  the  next  product ;  now  multiply 
the  second  ujyper  hy  units,  and  the  first  upper  by  tens, 
add  these  two  products  together,  setting  down  the  units 
figure  of  their  sum,  and  reserving  the  tens  to  carry,  as 
before  ;  now  multiply  the  third  upper  by  units  and 
the  second  upper  by  tens,  add  these  two  products  to- 
gether, setting  down  the  units  figure  of  their  sum  and 
reserving  the  tens  to  carry  as  usual ;  now  multiply  the 
third  ujyper  hy  tens,  add  in  the  reserved  figure,  if  there 
is  one,  and  set  down  the  whole  amount.  This  will  be 
the  complete  product. 

Remark. — One  of  the  principal  errors  with  the  be- 
ginner in  this  system  of  multiplication  is  neglecting 
to  add  in  the  reserved  figure.  The  student  must 
bear  in  mind  that  the  reserved  figure  is  added  on  to 
the  first  product  obtained  after  the  setting  down  of  a 
figure  in  the  complete  product. 


16  the  merchants  akd  mechanics* 

Example  1. — Explanation. 

Multiply  first  upper  by  units,  6X3  are  15,  123 
set  down  the  5,  reserve  the  one  to  carry  to         45 

the  next  product ;  now  multiply  second  upper 

by  units  and  first  upper  by  tens,  5X2  are  10  5535 
-f-l  are  11,  4X3  are  12,  add  these  products 
together  ll-)-12  are  23,  set  down  the  3,  reserve  the  2 
to  carry  ;  now  multiply  third  upper  by  units  and 
second  upper  by  tens,  add  these  two  products  to- 
gether, always  adding  on  the  reserved  figure  to  the 
first  product :  5X1  are  5-(-2  are  7,  4X2  are  8,  and 
7-|-8  are  15,  set  down  the  5,  reserve  the  1  ;  now  mul- 
tiply tliird  upper  by  tens,  and  set  down  the  whole 
amount :  4X1  are  4-)-l  are  5,  set  down  the  5.  This 
will  give  the  complete  product. 

Multiply  123  by  456  in  a  single  line. 

Here  the  first  and  second  places  are  found  123 
as  before;  for  the  third  add  6X1,  5X2,  4X3,         456 

with  the  2  you  had  to  carry,  making  30,  set 

down  0  and  carry  3,  then  drop  the  units'  56088 
place  and  multiply  the  hundreds  and  tens 
crosswise,  as  you  did  the  tens  and  units,  and  you 
find  the  thousand  figure  ;  then,  dropping  both  units 
and  tens,  multiply  the  4X1,  adding  the  one  you  car- 
ried, and  you  have  5,  which  completes  the  product. 
The  same  principle  may  be  extended  to  any  number 
of  places  ;  but  let  each  step  be  made  perfectly  fami- 
liar before  advancing  to  another.  Begin  with  two 
places,  then  take  three,  then  four,  but  always  prac- 
ticing some  time  on  each  number,  for  any  hesitation 
as  you  progress  will  confuse  you. 

N.  B. — The  following  mode  of  multiplying  num- 
bers will  only  apply  where  the  sum  of  the  two  last 
cr  unit  figures  equal  ten,  and  the  other  figures  in 
both  factors  are  the  same. 


COMMERCIAL   ARITHMETIC.  17 

Useful  and  Interesting  Contractions, 

when  the  multiplier  is  10,  100  or  1,  with  any  num- 
ber of  ciphers  annexed. 

Rule. — Annex  as  many  ciphers  to  the  multiplicand 
as  there  are  ciphers  in  the  multiplier;  the  number  so 
formed  will  be  the  ptoduct  required. 

Example  1.     Multiply  1854  by  10000. 

There  are/ow?'  ciphers  in  the  multiplier,  therefore 
annex ybwr  ciphers  to  the  rnultiplicand, 

7854X10000=78540000.     Product. 

To  multiply  any  number  composed  of 'two  figures 
by  11. 

Rule. —  Write  the  sum,  of  the  figures  between  them. 

2.  45X11=495.     Ans.     4+5=9. 

3.  62X11.  6+2=8',  placed  between  6  and  2= 
682.     Ans. 

Remark. — When  the  sum  of  the  two  figures  is  over 
9.  increase  the  left  hand  figure  by  the  one  to  carry 

4.  68X11-     Here,  6+8=14,  over  9.     Ans.  748- 

To  multiply  by  any  number  of  9s. 

Rule. — Annex  as  many  ciphers  to  the  multiplicand 
as  there  are  9s  in  the  multiplier,  and  subtract  the  mul- 
tiplicand. 

5.  Multiply  6472  by  999. 

Process.— 6472000— 6472=6465528.    Ans. 

To  write  down  the  square  of  any  number  of  9s 
at  ooot  without  multiplymg. 


18  THE    MERCHANTS    AND    MECHANICS* 

RuLE.-^-Trn'^e  down  as  many  9s,  less  one,  as  there 
are  9s  iVi  the  given  number,  an  8,  as  many  Os  as  9a|. 
and  a  1. 

6.  What  is  the  square  of  999  ?     Ans.  998001. 

Explanation. — As  there  are  three  9s  in  the  given 
number,  we  write  down  two  9s,  then  an  8,  then  two 
Os  and  a  1. 

T.  Square  9999999.     Ans.  99999980000001. 

To  multiply  any  number  by  two  figures  when  the 
unit  figure  is  1. 

KuLE. — Multiply  by  the  ten^s  figure,  and  set  the  pro- 
duct under  the  ten^s  figure  of  the  multiplicand  ;  then 
add. 

8.  Multiply  635t  by  41.  635t 

25428 


260637  Ans. 

Remark. — If  ciphers  intervene,  as  201,  40001,  etc., 
multiply  as  before,  but  set  the  product  as  many 
places  to  the  left  of  the  tens  as  th«re  are  ciphers. 

9.  Multiply  48546  by  3001.  48546 

145638 


145686546  Ans. 

To  multiply  when  there  are  ciphers  at  the  right  of 
f»iie  or  both  factors. 

Rule. — Multiply  the  significant  figures  together,  and 
annex  to  the  product  as  many  ciphers  as  there  are 
^^phori  on  ihe  right  of  both/aoior$. 


COMMERCIAT,    ARITHMETIC.  19 

Mental  Operations  in  Fractions. 
To  square  any  number  containing  J,  as  6J,  9J. 

Rule. — Multiply  the  whole  number  by  the  next  higher 
whole  number  and  annex  \  to  the  product. 

Example  1.  What  is  the  square  of  1 J  ?  Ans.  56J. 

"We  simply  say  1  times  8  are  56,  to  which  we 
add  J 

2.  What  will  9^  lbs.  beef  cost  at  9^  cts.  a  lb.  ? 

3.  What  will  \2\  yds.  tape  cost  at  12^  cts.  a  yd.  ? 

4.  What  will  5^  lbs.  nails  cost  at  b\  cts.  a  lb.  ? 

5.  What  will  11  ^  yds.  tape  cost  at  llj  cts.  a  yd.  ? 

6.  What  will  19|  bu.  bran  cost  at  19J  cts.  a  bu.? 

Reason. — We  multiply  the  whole  number  by  the 
next  higher  whole  number,  because  half  of  any  num- 
ber taken  twice  and  added  to  its  square  is  the  same 
as  to  multiply  the  given  number  by  one  more  than 
itself.  The  same  principle  will  multiply  any  two 
like  numbers  together,  when  the  sum  of  the  fractions 
is  one,  as  8 J  by  8§,  or  11|  by  llf,  etc.  It  is  obvi- 
ous that,  to  multiply  any  number  by  any  two  frac- 
tions whose  sum  is  one,  the  sum  of  the  products 
muat  be  the  original  number,  and  adding  the  number 
to  its  square  is  simply  to  multiply  it  by  one  more 
than  itself — for  instance,  to  multiply  ^J  by  7|  we 
simply  say  7  times  8  are  56,  and  then,  to  complete 
the  multiplication,  we  add,  of  course,  the  product  of 
the  fractions  (|  times  \  are  ^),  making  ^^-^  the 
answer. 

7.  Multiply  356000  by  2400.  356000 

2400 


1424 
712 


8^400000  A»8. 


20  THE    MERCHANTS    AND    MECHANICS* 

To  square  any  number  ending  in  5. 

Rule.—  Omit  the  5  and  multiply  the  number  as  it  will 
then  stand  by  the  next  higher  number,  and  annex  25  to 
the  product. 

8.  What  is  the  square  of  65  ?  Ans.  4225. 

Explanation. — We  simply  say  7  times  6  are  42^ 
and  annex  25. 

9.  What  is  the  square  of  85  ?  Ans.  T225. 

10.  Square  295.  Ans.  87025. 

Explanation. — Multiply  29  by  30  and  annex  25. 

To  square  any  number  containing  J. 

Rule. — Multiply  the  whole  number  by  the  next  higher 
whole  number  and  annex  \  to  the  product. 

11.  What  is  the  square  of  8|  ?  Ans.  72 J. 
We  simply  say  9  times  8  are  72  and  annex  J. 

12.  What  will  12^  pounds  beef  come  to  at  12 J 
cents  a  pound  ?  Ans.  1.56J. 

13.  What  will  GJ  pounds  spike  come  to  at  GJ 
cents  a  pound  ?  Ans.  42^. 

To  multiply  any  two  like  numbers  together  when 
the  sum  of  the  fractions  is  one. 

Rule. — Ilultij^ly  the  whole  number  by  the  next  higher 
whole  number,  and  to  the  product  add  the  product  of 
tht  fractions. 

Remark. — To  find  the  product  of  the  fractions 
multiply  the  numerators  together  for  a  new  numera- 
toi  and  the  denominators  for  a  new  denominator. 


COMMERCIAL    ARITHMETIC.  21 

14.  Multiply  6|  by  6|.  Ans.  42^\. 
Explanation. — Multiply  6,  the  whole  number,  by  t, 

the  next  higher  whole  number=42.  We  then  mul- 
tiply the  numerators  of  the  fractions,  2X^=6,  and 
the'  denominators,  5X^=25,  making  the  product 
•g'j,  which  we  add  to  the  product  of  the  whole  num- 
ber, 42, 

15.  Multiply  7J  by  7|.  Ans.  56|. 

16.  Multiply  114  by  II4,  Ans.  132-if. 

17.  Multiply  29J  by  29§.     Ans.  870f. 

To  multiply  any  two  like  numbers  together,  each 
of  which  has  a  fraction  with  a  like  denominator,  as 
3 J  by  5i,  or  6|  by  7|,  etc. 

Rule. — Add  to  the  midtiplicand  the  fraction  of  the 
m  idtiplier  and  midtiply  this  sum  by  the  ivhole  number; 
to  the  product  add  the  product  of  the  fractions. 

18.  Multiply  6i  by  5|.  Ans.  SS^?^. 

Tlie  sum  of  6J  and  |  is  7,  so  we  simply  say  6 
times  7  are  35  ;  to  this  we  add  the  product  of  the 
fractions,  j  times  J  are  ^=35^^.  Ans. 

19.  Multiply  9i  by  85.  Ans.  78^. 

The  sum  of  9J  and  f  is  9},  and  8  times  9|  are  78, 
to  which  add  the  product  of  the  fractions. 

WHERE   THE    SUM    OF   THE    FRACTIONS    IS    ONE. 

To  multiply  any  two  numbers  whose  difference  is 
one  and  the  sum  of  the  fractions  is  one. 

Rule. — Multiply  the  larger  number,  increased  by  one, 
by  the  smaller  number ;  then  square  the  fraction  of  the 
larger  number,  and  subtract  its  square  from  one. 


22  THE  MERCHANTS  AND  MECHANICS* 


PRACTICAL  EXAMPLES   FOR   BUSINESS 

MEN. 


1.  What  will  9J  lbs.  sugar  cost  at  8|  cts.  per  lb.  ? 

Here  we  multiply  9,  increased  by  1,  by  8,  9^ 

thus  :  8X10  are  80,  and  set  down  the  result ;  8| 

then  from    1    we   subtract  the   square  of  |,     

thus  :  J  squared  is  y'g-,  and  1  less  -^  is  -^f.  80J-| 

2.  What  will  8|  bu.  coal  cost  at  t|  cts.  a  bu.  ? 


Here  we  multiply  8,  increased  by  1,  by  T,  8| 

thus  :  1  times  9  are  63,  and  set  down  the  re-  t| 

suit  ;  then  from  1  we  subtract  the  square  of     

I,  thus  :  I  squared  is  f,  and  1,  less  f,  is  f.  63|- 

3.  What  will  11 /j  bu.  seed  cost  at  $10^3  a  bu.  ? 

Here  we  multiply  11,  increased  by  1,  by  10       11^% 
thus  :  10  times  12  are  120,  and  set  down  the       10|| 

result  ;  then  from  1   we  subtract  the  square  

of  -f^,  thus  :  -^^  squared  is  jj-^,  and  1  less  120|||- 
tI-^  is  iff 

4.  How  many  square  inches  in  a  floor  99|  in.  wide 
»r\d  98|  in.  long  ?     Ans.  9800f|-. 

Method    of  Qper'atlon. 

Example  First. 

Multiply  6J  by  6|  in  a  single  line. 

Here  we  add  6|-j-4>  which  gives  6J  ;  this  6| 

multiplied  by  the  6  in  the  multiplier,  6X6|,  6^ 

gives  39,  to  which  we  add  the  product  of  the     

fractions  ;  thus  ^Xi  gives  ■^\,  added  to  39  39^^ 
Bompletes  the  product. 


COMMERCIAL   ARITHMETIC.  28 

Example  Second. 

Multlx»ly  nibylljina  single  line. 

Here  we  would  add  lli+l»  which  gives  11 J 
12  ;  this  multiplied  by  the  11  in  the  multi-       11 1 

plier  gives  132,  to  which  we  add  the  product  

of  the  fractions  ;  thus  JXi  gives  ■^\,  which  132^ 
added  to  132  completes  the  product. 

Example  Third. 

Multiply  12Jbyl2Jina  single  line. 

Here  we  add  12|-|-J,  which  gives  13J  ;  this  12^ 

multiplied  by  the  12  in  the  multiplier,  12X13},  12| 

gives  159,  to  which   add  the  product  of  the  

fractions  ;    thus  JX2  gives  |,  which  added  to  159| 
159  completes  the  product. 

WHERE    THE    SUM    OF    THE    FRACTIONS    IS    ONE. 

To  multiply  any  two  like  numbers  together  when 
the  sum  of  the  fractions  is  one. 

Rule, — Multiply  the  whole  number  by  the  next  higher 
v:hole  number ,  after  which  add  the  product  of  the 
fractions. 

N.  B. — In  the  following  examples  the  product  of 
the  fractions  are  obtained ^rs/,  for  convenience  : 

Practical  Examples  for  Business  Men. 

Multiply  3|  by  3J  in  a  single  line. 

Here  we  multiply  JXI»  which  gives  -f^,  and  3| 

set  down  the  result ;  then  we  multiply  the  3  3| 

m   the  multiplicand,   increased   by  unity,  by     

the  3  in  the  multiplier,  3X4,  which  gives  12  12^^ 
and  completes  the  product. 


24  THE    MERCHANTS   AND    MECHANICS* 

Multiply  If  by  If  in  a  single  line. 

Here  we  multiply  fXI)  which  gives  ^V)  ^^^  *^| 

set  down  the  result  ;  then  we  multiply  the  *I  t| 

in  the  multiplicand,  increased  by  unity,  by     

the  1  in  the  multiplier,  tX^,  which  gives  56,  56^''^ 
and  completes  the  product. 

Multiply  11|  by  11§  in  a  single  line. 

Here  we  multiply  fXI;  which  gives  |-,  and  II | 

set  down  the  result ;  then  we  multiply  the  11  llf 

in  the  multiplicand,   increased  by  unity,  by  - — — 

the  11  in  the  multiplier,  11X12,  which  give&  132| 
132,  and  completes  the  product. 

Example  Fourth. 

Multiply  16|  by  16|  in  a  single  line. 

Here  we  multiply  JXI  which  gives  -|,  and  16| 

set  down  the  result ;  then  we  multiply  the  16  16| 

in  the  multiplicand,  increased  by  unity,  by     

the  16  in  the  multiplier,  16X1*7,  which  gives  2Y2| 
272,  and  completes  the  product. 

Example   Fifth. 

Multiply  29J  by  29|  in  a  single  line. 

Here  we  multiply  JXi  which  gives  J,  and  29J 

set  down  the  result  ;  then  we  multiply  the  29  29| 

in  the  multiplicand,  increased  by  unity,  by  the     

29  in  the  multiplier,  29X30,  which  gives  8tO,  870J 
and  completes  the  product. 

Example   Sixth. 
Multiply  999|  by  999^  in  a  single  line. 


COM>IERCIAL    ARITHMETIC.  ]  25 

Here  we  multiply  JXf*    which  gives  999| 

|-f ,  and  set  down  the  result  ;  then  we  999| 

multiply  the  999  in  the  multiplicand,  in-    

creased  by  unity,  by  the  999  in  the  mul-  990000f  J 
tiplier,  999X1000,  which  gives  999000, 
and  completes  the  product. 

Note. — The  system  of  multiplication  introduced 
in  the  preceding  examples  applies  to  all  numbers. 
Where  the  sum  of  the  fractions  is  one,  and  the  whole 
numbers  are  alike,  or  differ  by  one,  the  learner  is 
requested  to  study  well  these  useful  properties  of 
numbers. 

WHERE    THE    FRACTIONS    HAVE    A    LIKE    DENOMINATOR. 

To  multiply  any  two  like  numbers  together,  each 
of  which  has  a  fraction  with  a  like  denominator,  as 
43X^1,  or  lliXllf,  or  lOfXlO^  etc. 

Rule. — Add  to  the  multiplicand  the  fraction  of  tJie 
multiplier,  and  multiply  this  sum  by  the  whole  nurnber, 
after  which  add  the  product  of  the  fractions. 

Practical  Examples  for  Business  Men. 

N.  B. — In  the  following  example  the  sum  of  the 
fractions  is  one : 

1.  What  will  9J  lbs.  beef  cost  at  9i  cts.  a  lb.  ? 

The  sum  of  9J  and  J  is  10,  so  we  simply  9| 
pay  9  times  10  are  90  ;  then  we  add  the  ^ro-  9| 
duct  of  the  fractions,  \  times  j  are  ■^.  

N.  B. — In  the  following  example  the  sum  of  the 
fractions  is  less  than  one : 

i.  What  will  gj  yds.  tape  cost  at  8f  cts.  a  yd.  ? 


26  THE   MERCHAITTS    ANB   MECHANICS' 

The  sum  of  8 J  and  f-  is  8|,  so  we  simply         8| 
say  8  times  8|  are  tO  ;  then  we  add  the  pro-         8| 

duct  of  the  fractions,  f  times  J  are  ■^-^,  or  |.         

T0| 

N.  B. — In  the  following  example  the  sum  of  the 
fractions  is  greater  than  one : 

3.  ■\^hat  will  4 1  yds.  cloth  cost  at  $|  a  yd.  ? 

The  sum  of  4|  and  I  is  5  J,  so  we  simply         4| 
say  4  times   5 J  are  21;  then  we  add  the  pro-         4| 

duct  of  the  fractions,  |  times  |  are  -f^.  • 

21fi 

N.  B. — Where  the  fractions  have  different  denomi- 
nators reduce  them  to  a  common  denominator. 

RAPID    PROCESS    FOR   MULTIPLYING    MIXED    MUMBERS. 

A  valuable  and  useful  rule  for  the  accountant  in 
the  practical  calculations  of  the  counting  room. 

To  multiply  any  two  numbers  together,  each  of 
which  involves  the  fraction  | — as  t|X9,  etc. 

KuLE. — To  the  product  of  the  whole  numbers  add 
half  their  sum,  plus  J. 

Examples  for  Mental  Operations. 

1.  What  will  3i  doz.  eggs  cost  at  1|  cts.  a  doz.  ? 

Here  the  sum  of  7  and  3  is  10,  and  half  this         3J 
sum  is  5,  so  we  simply  say  t  times  3   are   21         1| 

and  5  are  26,  to  which  we  add  ^,  • 

261 

N.  B. — If  the  sum  be  an  odd  number  call 
it  one  less,  to  make  it  even,  and  in  such  cases  th© 
fraction  must  be  |. 


COMMERCIAL   ARITHMETIC.  27 

2.  What  will  11 J  lbs.  cheese  cost  at  9 J  cts.  a  lb.  ? 

3.  What  will  8|  j^ds.  tape  cost  at  15 J  cts.  a  yd.  ? 

4.  What  will  7^  lbs.  rice  cost  at  13|  cts.  a  1...  ? 
6.  What  will  10^  bu.  coal  cost  at  12|  cts.  a  bu.  ? 

Reason. — In  explaining  the  above  rule  we  add 
half  their  sum,  because  half  of  either  number  added 
to  half  the  other  would  be  half  their  sum,  and  we 
add  J  because  IXi  ^^  i-  ^^^^  same  principle  will 
multiply  any  two  numbers  together,  each  of  which 
has  the  same  fraction — for  instance,  if  the  fraction 
was  J  we  would  add  one  third  their  sum  ;  if  |,  we 
would  add  three  fourths  their  sum,  etc.  ;  and  then, 
to  complete  the  multiplication,  we  would  add,  of 
course,  the  product  of  the  fractions. 

6.  Multiply  4|  by  4|.     Ans.  21|^. 

The  sum  of  4|  and  |  is  6 J,  and  4  times  5  J  is  21; 
add  IXl=il     21fi-  Ans. 

To  multiply  any  two  numbers  together,  each  of 
which  involves  the  fraction  J. 

Rule. —  To  the  product  of  the  whole  numbeis  add 
half  their  sum,  j^lus  J. 

7.  Multiply  SJXH-     Ans.  26i. 

Solution. — The  sam  of  3  and  7  are  10,  and  one 
half  this  sum  is  5,  so  we  say  7  times  3  are  21  and  5 
are  26,  to  which  we  annex  J.     26^  Ans. 

8.  What  will  71  lbs.  cheese  cost  at  13i  cts.  a  lb.? 
Au.s.  $1.01^. 

Remark. — If  the  eum  be  an  odd  number  call  it 
one  less,  to  make  it  even ;  in  which  case  the  fraction 
must  be  }. 


9.  What  will  8|  lbs.  of  sugar  cost  at  15|  cts.  a 
lb.?     Ans.  $1.31|. 

Here,  8-1-15=23,  being  an  odd  number,  we  make 
It  one  less,  22,  one  half  of  which  is  11,  Then,  8 
times  15  are  120,  and  11  are  131,  to  which  we  add  |. 

The  same  principle  will  multiply  any  two  numbers 
together,  each  of  which  has  the  same  fraction.  For 
instance,  if  the  fraction  was  |-,  Ave  would  add  one 
fifth  their  sum  ;  if  j,  we  would  add  three  fourths 
their  sum  ;  if  |,  add  two  thirds  their  sum,  etc.,  after 
which,  of  course,  add  the  product  of  their  fractions, 

10.  Multiply  8|X^|.     Ans.  66f 

The  sum  of  8  and  t  are  15,  two  thirds  of  which  is 
10.  We  then  sa,j,  8  times  1  are  56  and  10  make  66, 
and  add  fXl^i- 

Business  men  generally,  in  multiplying,  only  care 
about  havmg  the  answer  correct  to  the  nearest  cent 
— that  is,  they  disregard  the  fraction.  When  it  is  a 
half  cent  or  more  they  call  it  another  cent  ;  if  less 
than  half  a  cent,  they  drop  it.  Therefore,  to  multi- 
ply any  two  numbers  to  the  nearest  unit  we  give 
the  following 

General  Rule. 

I.  MultiiJly  the  whole  number  in  the  multiplicand  by 
Refraction  in  the  multiplier  to  the  nearest  unit. 

11.  Multiply  the  whole  number  in  the  multiplier  by 
ihe  fraction  in  the  multiplicand  to  the  nearest  unit. 

III.  MultiiDly  the  whole  numbers  together,  and  add 
the  three  products  in  your  mind  as  you  proceed. 

Remark. — This  rule  is  very  simple  and  true,  and 
there  being  no  such  thing  as  a  fraction  to  add  in, 
there  is  scarcely  any  liability  to  error  or  mistake. 


COMMERCIAL    ARITHMETIC.  29 

The  work  can  generally  be  done  mentally,  for  only 
easy  fractions  actually  occur  in  business. 

11.  Multiply  9|  by  8J.     Ans.  17. 

Solution. — J  of  9  is  nearer  2  than  3,  and  \  of  8  is 
nearer  3  than  2.  Make  the  nearest  whole  number 
the  quotient.  2  and  3  are  5,  so  we  simply  say  8 
times  9  are  72  and  5  are  *IT.  Ans. 

12.  Multiply  11§  by  7f     Ans.  91. 

Here,  ^  of  11  to  the  nearest  unit  is  9,  and  |  of  7  to 
the  nearest  unit  is  5  ;  then  9-[-5=14,  so  we  say  7 
times  11  are  77,  and  14  are  91.  Ans. 

The  component  factors  of  a  number  are  such  fac- 
tors as  multiplied  together  will  produce  that  num- 
ber— thus,  3  and  4  are  compound  factors  of  12, 
because  3X4=12  ;  also,  2,  2  and  3,  because  2X2X 
3=12  ;  also,  6X2=12. 

13.  Multiply  128f  by  25,  by  business  128| 
method.  25 

Here  f  of  25  to  the  nearest  unit  is  17,  so  we  . 

simply  say,  25  times  128  are  3200,  and  17  are  3217 
3217,  the  answer. 

Practical  Examples  for  Business  Men. 

1.  What  is  the  cost  of  17^  lbs.  sugar  at  18|  cts. 
per  lb,  ? 

Here  |  of  17  to  the  nearest  unit  is  13,  and  17  J 
\  of  18  is  9,  13  plus  9  is  22,  so  we  simply  say     18j 

18  times  17  are  306,  and  22  are  328,  tho  an 

swer.  '  $3.28 

2.  What  Is  the  cost  of  11  lbs.  5  bz.  of  butter  rJ 
33^  cts.  per  lb.  ? 


so  THE    MERCHANTS    AND    MECHANICS' 

Here  ^  of  11  to  the  nearest  unit  is  4,  and  H  A" 
^  of  33  to  the  nearest  unit  is  10,  then  4         33-^ 

-j-lO  is  14,  so  we  simply  say  33  times  11  

are  363,  and  14  are  37 1,  the  answer.  $3.7 T 

3.  What  is  the  cost  of  17  doz.  and  9  eggs,  at  12| 
cts.  per  doz.  ? 

Here  |  of  17  to  the  nearest  unit  is  9,  and  1*1 -^\ 
-,9^  of  12  is  9  ;  then  9+9=18,  so  we  simply  12^ 

say  12  times  17   are   204,  and  18  are   222, 

the  answer.  $2.22 

4.  What  will  be  the  cost  of  15|  yds.  calico  at  12| 
cts.  per  yd.?     Ans.  $1.97. 

N.  B. — To  multiply  by  aliquot  parts  of  100  see 
page  44. 

Rapid  Process  of  Marking  Goods. 

A  VALUABLE  HINT  TO  MERCHANTS  AND  ALL  RETAIL  DEALERS 
IN  FOREIGN  AND  DOMESTIC  DRY  GOODS. 

Retail  merchants,  in  buying  goods  by  wholesale, 
buy  a  great  many  articles  by  the  dozen,  such  as 
boots  and  shoes,  hats  and  caps,  and  notions  of  vari- 
ous kinds.  Now  the  merchant,  in  buying,  for  in- 
stance, a  dozen  hats,  knows  exactly  what  one  of 
those  hats  will  retail  for  in  the  market  where  he 
deals  ;  and,  unless  he  is  a  good  accountant,  it  will 
often  take  him  some  time  to  determine  whether  he 
can  afford  to  purchase  the  dozen  hats  and  make  a 
living  profit  in  selling  them  by  the  single  hat ;  and; 
in  buying  his  goods  by  auction,  as  the  merchant 
often  does,  he  has  not  time  to  make  the  calculation 
before  the  goods  are  cried  off ;  be  therefore  loses 
the  chance  of  making  good  bargains  by  being  afraid 
to  bid  at  random,  or  if  he  bids,  and  the  gooda  are 


COJniERCIAT,    ARITHMETIC.  31 

cried  off,  he  may  have  made  a  poor  bargain  by  bid- 
ding thus  at  a  venkire.  It  then  becomes  a  useful 
and  practical  problem  to  determine  instantly  what 
per  cent,  he  would  gain  if  he  retailed  the  hats  at  a 
certain  price. 

To  tell  what  an  article  should  retail  for  to  make 
a  profit  of  20  per  cent.  : 

Rule, — Divide  what  the  articles  cost  per  dozen  by  10, 
which  is  done  by  removing  the  decimal  point  one  place 
to  the  left. 

For  instance,  if  hats  cost  $17.50  per  dozen,  remove 
the  decimal  point  one  place  to  the  left,  making  $1.75, 
what  they  should  be  sold  for  apiece  to  gain  20  per 
cent,  on  the  cost.  If  they  cost  $31.00  per  dozen  they 
should  be  sold  at  $3.10  apiece,  etc.  We  take  20  per 
cent,  as  the  basis  for  the  following  reasons,  viz.,  be- 
cause we  can  determine  instantly,  by  simply  remov- 
ing the  decimal  point  without  changing  a  figure  ; 
and,  if  the  goods  would  not  bring  at  least  20  per 
cent,  profit  in  the  home  market,  the  merchant  could 
not  afford  to  purchase,  and  would  look  for  goods  at 
lower  figures. 

Reason. — The  reason  for  the  above  rule  is  obvious 
— for  if  we  divide  the  cost  of  a  dozen  by  12  we  have 
the  cost  of  a  single  article  ;  then,  if  we  wish  to  make 
20  per  cent,  on  the  cost  (cost  being  \  or  |),  we  add 
the  20  per  cent.,  which  is  ^,  to  the  4,  making  f  or  ||^; 
then,  as  we  multiply  the  cost,  divided  by  12,  by  the 
\^  to  find  at  what  price  one  must  be  sold  to  gain  20 
per  cent.,  it  is  evident  that  the  12s  will  cancel,  and 
leave  the  cost  of  a  dozen  to  be  divided  by  10,  which 
is  done  by  removing  the  decimal  point  one  place  to 
the  left. 


8^  THE  MERCHANTS  AKD  MECHANICS* 

1.  If  I  buy  2  doz.  caps,  at  $7.50  per  doz.,  what  shall 
I  retail  them  at  to  make  20%  ?  Ans.  75  cts. 

2.  When  a  merchant  retails  a  vest  at  $4.50  and 
makes  20%  what  did  he  pay  per  doz.  ?  Ans.  $45. 

3.  At  what  price  should  I  retail  a  pair  of  boots, 
that  cost  $85  per  doz.,  to  make  20%  ?     Ans.  $8.50. 

Rapid  Process  of  Marking  Goods  at  Different 
per  cents. 

Now,  as  removing  one  decimal  point  one  place  to 
the  left,  on  the  cost  of  a  dozen  articles,  gives  the 
selling  price  of  a  single  one  with  20  per  cent,  added 
to  the  cost,  and  as  the  cost  of  any  article  is  100 
per  cent.,  it  is  obvious  that  the  selling  price  would 
be  20  per  cent,  more,  or  120  per  cent.  ;  hence,  to  find 
50  per  cent,  profit,  which  would  make  the  selling 
price  150  per  cent,,  we  would  first  find  120  percent., 
then  add  30  per  cent,  by  increasing  it  one  fourth 
itself.  To  make  40  per  cent.,  add  20  per  cent.,  by 
increasing  it  one  sixth  itself;  for  35  per  cent,  in- 
crease it  one  eighth  itself,  etc.  Hence,  to  mark  an 
article  at  any  per  cent,  profit  we  have  the  following 

General  Rule. 

First  find  20  per  cent,  profit  by  removing  the  deci- 
mal jyoint  one  place  to  the  left  on  the  price  the  articles 
cost  a  dozen;  then,  as  '2,0  per  cent. profit  is  120  per 
cent.,  add  to  or  subtractfrom  this  amount  the  fractional 
part  that  the  required  per  cent,  added  to  100  is  more  or 
less  than  120. 

Merchants,  in  marking  goods,  generally  take  a  per 
cent,  that  is  an  aliquot  part  of  100,  as  25%,  3S|%, 
50%,  etc.  The  reason  they  do  this  is  because  it 
xaakes  it  much  easier  to  add  such  a  j^t-  cent,  t^  tl\« 


COMMERCIAL    ARITHMETIC. 


33 


cost.  For  instance,  a  merchant  could  mart*  almost 
a  dozen  articles  at  50  per  cent,  profit  in  the  time 
it  would  take  him  to  mark  a  single  one  at  49  per 
cent.  For  the  benefit  of  the  student,  and  for  the 
convenience  of  business  men  in  marking  goods,  we 
have  arranged  the  following 

Table 

FOR  MARKING  ALL  ARTICLES  BOUGHT  BY  THE  DOZEN. 

N.  B. — Most  of  these  are  used  in  business. 
To  make  20%  remove  the  point  one  place  to  the  left 


*      80% 

'     and  add  i  itself. 

'      60% 

'         "      "    1     " 

'      50% 

u        u      J       a 

'      44% 

'             "        "      \      " 

'      40% 

'             "        "      i      " 

'      37^% 

'             "        ".    \      " 

'      35% 

'             -        -      1      - 

'  m% 

'             "        "      i      " 

'     32% 

'       "    "  iV  " 

'      30% 

'       "    "  iV  "   ■ 

'      28% 

'     "   "  IV  " 

'      26% 

'             "        "    ^G     " 

'      25% 

'             "        "    ^     " 

'     12|% 

*    subtract  -^    ** 

'     18§% 

■     "     A  ■' 

'  m% 

'    .-"  .  ^.  ■'  , 

If  I  buy  1  doz.  shirts  for  $28.00  what  shall  I  re- 
tail  them  for  to  make  50%  ?     Ans.  $3  50. 

Explanation. — Remove  the  point  one  place  to  the 
left  and  add  on  J  itself. 

WHERE  THE  MULTIPLIER  IS  AN  ALIQUOT  PART  OF  100. 

Merchants,  in  selling  goods,  generally  make  the 
price  of  an  article  some  aliquot  part  of  lot),  as  in 


S4  THE    MERCHANTS    AND    MECHANICS^ 

selling  sugar  at  12J  cents  a  pound,  or  8  pounds  for 
1  dollar,  or  in  selling  calico  for  16|  cents  a  yard,  or 
6  yards  for  1  dollar,  etc.  And  to  become  familiar 
with  all  the  aliquot  parts  of  100,  so  that  you  can 
apply  them  readily  when  occasion  requires,  is,  per- 
haps, the  most  useful,  and,  at  the  same  time,  one  of 
the  easiest  arrived  at  of  all  the  computations  the 
accountant  must  perform  in  the  practical  calcula- 
tions of  the  counting-room. 

Table  of  the  Aliquot  parts  of  lOO  and  1,000. 
N.  B. — Most  of  these  are  used  in  business. 

12i  is  I  part  of  100.  8  J  is^  part  of  100. 

25    is  I  or  4  of  100.  16§  isy^^  or  |  of  100. 


37i  is  f  part  of  100. 
50^  is  f  or  1  of  100. 

331  isy^  or  i  of  100. 
66§  is  -j^  or  2  of  100. 

621  is  J  part  of  100. 

83|  isif  orfof  100. 

75    is  1  or  j  of  100. 
87 J  is  1  part  of  100. 
64  is -jJ^  part  of  100. 
185isy3_part  of  100. 
314isy\part  of  100. 

125  is  i  part  of  1,000. 
250  is  f  or  i  of  1,000. 
375  ig  3  part  of  1,000. 
625  is  1  part  of  1,000. 
875    is  1  part  of  1,000. 

To  multiply  by  an  aliquot  part  of  100. 

Rule. — Add  two  ciphers  to  the  multiplicand,  then 
take  such  jyart  of  it  as  the  multiplier  is  part  of  100. 

N.  B. — If  the  multiplicand  is  a  mixed  number  re- 
dace  the  fraction  to  a  decimal  of  two  places  before 
dividing. 

To  multiply  by  an  aliquot  part  of  100. 

"RvLE. — Add  two  ciphers  to  the  multiplicand,  then 
take  such  part  of  it  as  the  multiplier  is  paH  of  100. 


COMMKKCIAL    ARITHMETIC.  35 

To  multiply  by  12J,  add  two  ciphers  and  divide 
by  8. 

1.  Multiply  2592  by  121.     Product,  32400. 
8)259200 


32400 


To  multiply  by  37^,  annex  two  ciphers   and  take 
i  j)f  it. 

J.  Multiply  1432  by  37^.     Product,  53700. 
8)14300 


17900 
8 

53700 


To  multiply  by  6|,  add  two  ciphers  and  divide  by 
15  ;  or  add  one  cipher /ind  multiply  by  |. 

3.  Multiply  6525  by  6§.     Product,  43500. 
15)652500  or,  3)65250 


43500  21750 

2 


43500 


To  multiply  by  87^,  add  two  ciphers,  divide  by 
8  and  subtract  the  quotient,  or  multiply  the  quo- 
tient by  7. 

4.  Multiply  6768  by  87i.     Product,  592200. 


36  THE    MERCHANTS    AND.  MECHANICS* 


8)676800  .  or,  8)676800 

84600  

84600 


692200  7 


592200 


To   multiply  bj    75,  add   two   ciphers   and   sub- 
tract J. 

5.  Multiply  4968  by  75.     Ans.  372600. 

4)496800 
124200 


372600 


To  multiply  by  125,  add  three  ciphers  and  divide 
by  8. 

6.  Multiply  3467  by  125.     Ans.  433375. 

8)3467000 


43^375 


To  multiply  by  875,   add  three  ciphers  and  sub 


tract  |. 


7.  Multiply  25136  by  875.     Ans.  21994000. 

8)25136000 
3142000 


21994000 


Every  fact  of  this  kind,  though  extremely  simple, 
will  be  found  very  useful  many  times,  and  should  be 
known  by  all  who  seek  to  be  skilful  in  figures. 


COMMERCIAL    ARITHMETIC.  37 


THE  GREAT  SECRET  OF  MATHEMATICS 
REVEALED. 

Nearly  every  problem  in  mathematics,  of  what- 
ever name,  will  come  under  one  of  the  three  follow- 
)ing  heads,  and  can  be  resolved  by  the  appropriate 
rule  belonging  to  them.  For  this  reason  they  should 
be  well  committed  to  memory  and  carefully  studied 
until  the  learner  becomes  perfectly  familiar  with  each 
and  all  of  them.  We  will  first  give  them  all,  after 
which  we  will  exemplify  each  one  of  them  sepa- 
rately : 

1.  The  price  of  one  and  the  quantity  being  given, 
to  find  the  cost  of  the  quantity. 

Rule. — Multiply  the  price  of  oj^e  by  the  quantity. 

2.  The  COST  and  the  quanttfy  being  given,  .to  find 
the  price  of  one. 

Rule. — Divide  the  cost  by  the  quantity. 

3.  The  price  of  one  and  the  cost  of  a  quantity  being 
given,  to  find  the  quantity. 

Rule. — Divide  the  cost  of  the  quantity  by  the  price 

of  OKE. 

Case  I. — The  price  of  one  and  the  quantity  being 
giv6n,  to  find  the  cost  of  the  quantity. 

Example  1. — If  one  acre  of  land  cost  $15,  what  will 
60  acres  cost  ? 

Multiply  the  price  of  one  acre,  $15,  br  tb»  quan- 
tity, 60  acres  :     15X60=$900.  Ans. 


38  THE    MERCHANTS    AND    MECHANICS* 

2.  What  commission  must  be  paid  for  collecting 
$11380  at  ^  per  cent.  ? 

Multiply  the  price  of  collecting  one  dollar  (3| 
cents)  by  the  quantity,  $17380  :  .OSIXIl'^SBO^^ 
$608.30.   Ans. 

3.  A  broker  negotiates  a  bill  of  exchange  of  $2890 
for  4^  per  cent,  commission.  How  much  is  his  bro- 
kerage  ? 

Multiply  the  price  of  negotiating  one  dollar  (^  of 
a  cent)  by  the  quantity,  $2890.     $23.12.    Ans. 

4.  If  the  stock  of  an  insurance  company  sells  at  5 
per  cent,  below  par,  what  will  $1200  of  the  stock 
cost? 

If  the  stock  was  at  par  one  dollar's  worth  of  stock 
would  be  worth  $1,  but  as  it  is  5  per  cent,  below 
par,  one  dollar  of  stock  is  only  worth  95  cents;  there- 
fore, multiply  95  cents,  the  price  of  one,  by  the 
quantity,  $1200.     $1140.    Ans. 

5.  What  will  cost  $5364  stock  in  the  Bank  of  Or- 
leans, at  9  per  cent,  above  par  ? 

Since  it  is  above  par  one  dollar  of  stock  is  worth 
$1.09;  therefore,  multiply  $1.09  by  $5364.  $5846.76. 
Ans. 

6.  What  is  the  interest  on  $512  for  three  years,  at 
7  per  cent.  ? 

The  price  of  one  dollar  for  three  years  at  7  per 
cent,  is  21  cents  ;  therefore,  multiply  the  price,  21 
pents,  by  $512.     $107.52.    Ans. 

7.  What  premium  must  be  paid  for  $4572.80  insu- 
rance at  2 1  per  cent.  ? 


COMMERCIAL    ARITHMETIC.  39 

MulUplv  2J  cents,  the  price  of  one  dollar,  by  the 
quantity,  $4572.80.     $114.32.  Ans. 

Case  II. — The  cost  and  the  quantity  being  given, 
to  find  the  price  of  one. 

Example  1. — If  25  acres  of  land  cost  $115,  what 
will  one  acre  cost  ? 

Divide  the  cost  of  the  quantity,  $175,  by  the  quan- 
tity. 25.     $t.  Ans. 

2.  A  man  Slaving  $125  lost  $5.     What  per  cent, 
of  his  money  did  he  lose  ? 

Divide  the  cost  of  the  quantity,  $5,  by  the  quan- 
tity, $125  :     $5-f-$125=.04  per  cent.  Ans. 

3.  I  lent  $450  for  one  year,  and  received  for  inter- 
est $31.50.     What  was  the  rate  per  cent.  ? 

Bivide  the  cost  of  the   quantitv,   $31.50,  by  the 
quantity,  $450  :     $31.50-f-$450=.6T  per  cent.  Ans. 

Case  III. — The  price  of  one  and  the  cost  of  a  quan- 
tity being  given,  to  fine  the  quantity. 

« 

Example  1. — At  $6  a  barrel  for  flour  how  many  bar- 
rels can  be  bought  for  $840  ? 

Divide  the  cost  of  the  quantity  by  the  price  of 
one  :     $840^$6«=140.  Ans. 

2.  A  man  lost  $5,  which  was  4  per  cent,  of  all  the 
money  he  had.     How  much  had  he  at  first  ? 

Divide  the  cost  of  the  quantitv,  $5,   bv  the  price 
of  one,  .04:     $5-^.04=$12n. 

3.  What  amount  of  stock  can  be  bought  for  $9682, 
allowing  3  per  cent,  brokerage  ? 


40  THE    MERCHANTS    AND    MECHANICS* 

Every  dollar's  worth  of  stock  costs  $1.03  ;  there- 
fore divide  the  cost  of  the  quantity,  $9682,  by  the 
price  of  one  :     $9682-^$1.03=$9400.  Ans. 

4.  What  principal,  in  2  years  6  months,  at  1  per 
cent.,  will  amount  to  $88,125  ? 

The  price  of  $1  for  2  years  6  months  will  be  $1,175; 
therefore  divide  the  cost  of  the  quantity.  $88,125,  by 
$1,175.     $88.125-^$1.175=$t5.  Ans. 

5.  In  what  time  will  $360  gain  $86.40  interest  at 
6  per  cent.  ? 

The  price  of  $360  for  one  year  at  6  per  cent,  will  be 
$21.60  ;  therefore,  divide  the  cost,  $86.40,  by  the 
price  of  one  year,  which  will  give  the  number  of 
years.     $86.40-f-$21.60=4  years.  Ans. 

INTEREST 

is  a  sum  paid  for  the  use  of  money. 

Principal  is  the  sum  for  the  use  of  which  interest 
is  paid. 

Amount  is  the  sum  of  the  principal  and  interest. 

Rate  per  cent.,  commonly  expressed  decimally  as 
hundredths,  is  the  sum  per  cent,  paid  for  the  use  of 
one  dollar  annually. 

.  Simple  IntereM  is  the  sum  paid  for  the  use  of  the 
principal  only  during  the  whole  time  of  the  loan. 

Legal  Intered  is  the  rate  per  cent,  established  by 
law. 

Usury  is  illegal  interest,  or  a  greater  per  c«;nt  than 
the  legal  rate. 


COMMERCIAL    ARITHMETIC.  41 

It  is  contended  by  many  statesmen  that  the  rate 
of  interest  should  not  be  established  by  statute,  but 
that  money  is  only  a  commodity  that,  like  every 
other  article  of  traffic,  should  be  governed  by  the 
law  of  supply  and  demand.  If  money  is  scarce  the 
rate  would  be  high  ;  if  plenty,  then  low.  But  as 
banks  and  other  great  monied  institutions  have  the 
power,  to  a  great  extent,  of  controlling  the  quantity 
of  money  in  the  market,  thereby  oppressing  the 
great  majority  of  the  people,  and  taking  advantage 
of  the  times  of  scarcity,  public  opinion,  at  least,  has 
established  the  law  of  usury. 

The  Jews  appear  to  be  about"  the  first  nation 
among  whom  we  find  a  distinct  class  called  "  money 
changers"  ©r  "  lenders,"  and  among  them  we  find  a 
law  existed  that  they  should  not  take  interest  of 
their  brethren,  though  they  were  permitted  to  take 
it  of  foreigners.  "  Thou  shalt  not  lend  upon  usury 
to  thy  brother — usury  of  money,  usury  of  victuals, 
usury  of  anything  that  is  lent  upon  usury  ;  unto  a 
stranger  thou  mayest  lend  upon  usury  ;  but  unto 
thy  brother  thou  shalt  not  lend  upon  usury." — (Deut. 
xxiii,  19,  20.)  After  the  dispersion  of  the  Jews  they 
wandered  through  the  earth — but  they  yet  remain  a 
distinct  people,  mixing  but  not  becoming  assimilated 
with  the  people  among  whom  they  reside.  Still 
looking  forward  to  the  period  when  they  shall  return 
to  the  promised  land,  they  seldom  engage  in  perma- 
nent business,  but  pursue  traffic,  especially  dealing 
in  money  ;  and  if  their  national  policy  forbids  their 
taking  interest  of  each  other,  they  show  no  back- 
wardness in  taking  it  unsparingly  of  the  rest  of  man- 
kind. For  ages  they  have  been  the  money  lenders 
of  Europe,  and  we  may  safely  attribute  to  this  cir- 
cumstance the  prejudice,  in  some  measure,  that  still 
exists,  even  in  our  own  country,  against  such  as  pur* 


42  THE    MERCHANTS    AXD    MECHANICS^  " 

sue  this  business  as  a  profession.  The  prejudice  of 
the  Christian  against  the  Jew  has  been  transferred 
to  his  occupation,  and  from  the  days  of  the  inexora- 
ble Shylock,  contending  for  his  pound  of  flesh,  down 
to  the  present  time,  the  grasping  money  lender,  no 
less  than  the  grinding  dealer  in  other  matters,  has 
been  sneeringly  called  a  Jew, 

The  rate  of  legal  interest  varies  in  different  States, 
and  we  subjoin  a  table  giving  the  legal  rate  as  well 
as  that  allowed  by  contract.  When  the  rate  per 
cent,  is  not  specified  in  accounts,  notes,  mortgages, 
contracts,  etc ,  the  legal  rate  is  always  understood. 


COMMERCIAL   ARITHMETIC. 


it 


Rates  of  Interest  and  Statute  Limitations  in 
the  United  States. 


States. 


Alabama . . . 
Arkansas . . . 
California.. 
Connecticut 
Delaware  . . 

Florida 

Greor^ia 

Illinois 

Indiana 

Iowa 

Kentucky.. 
Louisiana.. 

Maine 

Marjiand  . . 

Mass 

Michigan... 
Minnesota.. 
MlBsissippi. 
Misnouri. .. 
N.  Hampsh'e 
New  Jersey 
New  York  . 

N.  Carolina. 

Ohio 

Pennsylvania 
Rhode  Island 
S.  Carolina 
Tennessee. 

Texas 

Vermont.. 
Virginia... 
WiBconsin. 


10 
free 
10 
10 


12 


Penalty  for  Usury. 


Forfeiture  of  entire  interest. 
Usurious  contracts  void 


Forfeiture  of  entire  interest 

Forfeiture  of  entire  principal 

Forfeiture  of  entire  interest 

Forfeiture  of  excess  of  interest.. 

Forfeiture  of  entire  interest 

Usurious  interest  recoverable... 
Usurious  interest  recoverable... 

Usurious  excess  void 

Forfeiture  of  entire  interest 

Usurious  excess  void 

Forfeit  of  usury 

Forfeit  threefold  usurious  int.  taken 
Usurious  excess  void 


Forfeiture  of  interest 

Forfeiture  of  interest 

Forfeit  threefold  usurious  int.  taken 

Contract  void 

Contract  void.     Fine  not  over  $100, 

impris'mt  not  over  C  mos.,  or  both 

Forfeit  double  the  debt. 

Usurious  excess  void 

Forfeit  entire  principal  and  interest 

Usuiious  excess  void 

Forfeit  entire  interest 

Finn  at  least  $10 

Forfeit  entire  interest : . . . . 

Usurious  excess  void 

Contract  void 

Forfeit  entire  debt 


Btat.  Lim. 


12 


20 


To  find  the  interest  if  the  time  consists  of  years. 

Rule. — MuUiply  the  principal  by  the  rate  per  cerU,^ 
and  that  product  by  the  number  of  years. 


44 


THE    MERCHANTS   AND    MECHANICS' 


Example  1. — What  is  the  interest   of  $150  for  3 
years,  at  8  per  cent.  ? 

$150 
.08 

12.00 
3 


$36.00  Ans. 

The  decimal  for  8  per  cent,  is  .08.  There  being 
two  places  of  decimals  in  the  multiplier  we  point  off 
two  places  in  the  product. 

To  find  the  interest  when  the  time  consists  of 
years  and  months. 

KuLE. — Reduce  the  time  to  months.  Multiply  the 
principal  by  the  rate  per  cent.,  divide  the  product  by 
12,  and  the  quotient  multiplied  by  the  number  of 
months  will  be  the  interest  required. 

Or,  by  Cancellation. — Place  the  principal,  rate  and 
time  in  months  on  the  right  of  the  line,  and  12  on  the 
left,  then  cancel. 

2.  Find  the  interest  of  $240  for  3  years  and  T 
months,  at  t  per  cent. 

Principal,         $240 
Eate,  .07 


12)16.80 


BY  CANCELLATION. 


1.40 

$^/f0     20 

yrs+7  mos.      31 

Xt 

7 

31 

1.40 

20X7X31==$43.4(?    Ans. 

4.20 

$43.40  Ans. 


COMMERCIAL    ARITHMETIC.  45 


SIMPLE   INTEREST    BY  CANCELLATION. 

Rule. — Place  the  principal,  time  and  rate  per  cent, 
on  the  right  hand  side  of  the  line.  If  the  time  consists 
of  years  and  months,  reduce  them  to  months,  and  jylace 
12  {the  number  of  months  in  a  year)  on  the  left  hand 
side  of  the  line.  Should  the  time  consist  of  months 
and  days,  reduce  them  to  days,  or  decimal  parts  of  a 
■month.  If  reduced  to  days,  place  ^Q  on  the  left.  If  to 
decimal  parts  of  a  month,  place  12  only  as  before. 

Point  off  tivo  decimal  places  ivhen  the  time  is  in 
months,  and  three  decimal  places  when  the  time  is  in 
days. 

Note. — If  the  principal  contains  cents,  point  off 
four  decimal  places  when  the  time  is  in  months,  and 
five  decimal  places  when  the  time  is  in  days. 

(We  place  36  on  the  left,  because  there  are  360 
interest  days  in  a  year.  Custom  has  made  this 
lawful.) 

Example  1. — What  is  the  interest  on  $60  for  lit 
days,  at  6  per  cent.  ? 

OPERATION. 

Here  117X0 
must  be  the    $0 
answer. 


00  Both  6s  on  the  right 

0  cancel  36  on  the 

in  left,  and  we  have 

nothing  to  divide 


$1,170  Ans.  by. 

In  this  case  we  point  off  three  decimal  places,  be- 
cause the  time  is  in  days.  If  the  time  had  been  117 
months  we  would  have  pointed  off  but  two  decimal 
places. 

Example  2.— What  is  the  interest  of  $96.50  for  90 
days,  at  6  per  cent.  ? 


46 


THE   MERCHANTS   AND   MECHANICS^ 


OPERATION. 


$—H 


96.50 
00 — 15 
0 


9650 
15 


1.44.150  Ans. 


Now  cancel  6  in  36  and  the  quotient  6  into  90, 
and  we  have  no  divisor  left.  Hence,  15X96.50  must 
be  the  answer. 

Note. — As  there  are  cents  in  the  principal  we 
point  off  five  decimals — three  for  days  and  two  for 
cents.  Pay  no  attention  to  the  decimal  point  until 
the  close  of  the  operation. 

Example  3. — What  is  the  interest  of  $480  for  361 
days  at  6  per  cent.  ? 

4^0—80  361 

0— $0     361  80 

0  

$28,880  Ans. 

Now  cancel  6  in  36  and  the  quotient  6  into  480, 
and  we  have  no  divisor  left.  Hence,  80X^61  must 
be  the  answer. 


Example  4. — What  is  the 
months,  at  7  per  cent.  ? 


interest  of  $t20  for   9 

60 
9 

540 

1 


$3t.80  Ans. 
Now  cancel  12  in  *I20  and  there  is  nothing  left  to 
divide  by.     Hence,  eOX^X*^  must  be  the  answer. 


COMMERCIAL    ARITHMETia  4t 

N.  B. — When  interest  is  required  on  any  sum  for 
days  only,  it  is  a  universal  custom  to  consider  30 
days  a  month,  and  12  months  a  year  ;  and,  as  the 
unit  of  time  is  a  year,  the  interest  of  any  sum  for 
one  day  is  -g-J-g-  what  it  would  be  for  a  year.  For  2 
days,  y|^,  etc.  ;  hence,  if  we  multiply  by  the  days, 
we  must  divide  by  360,  or  divide  by  36  and  save 
labor.  -  The  old  form  of  this  method  was  to  place 
360,  or  12  and  30,  on  the  left  of  the  line,  but  using 
36  is  much  shorter. 

WHEN  THE  DATES  ARE  NOT  DIVISIBLE  BY  THREE. 

XoTE. — When  the  time  consists  of  months  and 
days,  and  the  days  are  not  divisible  by  three,  reduce 
the  time  to  days. 

Example  5. — What  is  the  interest  of  i960  for  11 
months  and  20  days  at  6  per  cent.  ? 

Months.        Days. 

OPERATION.  11     20=350  days. 

000—160  350 

0—36     350  160 

6  

$56,000 

Now  cancel  6  in  36  and  the  quotient  6  into  960, 
and  we  have  no  divisor  left.  Hence,  160X^50  must 
be  the  answer. 

Example  6.— What  is  the  interest  of  $173  for  8 
months  and  16  days,  at  9  per  cent  ? 

Month*.       Days. 

OPERATION.  8     16=256  days. 

173  173 

ii—H     0  64 

m-di         

|11.0t2  Am. 


48         .  THE    MERCHANTS    AND    MECHANICS^ 

Now  cancel  9  in  36  and  the  quotient  4  into  256, 
and  we  have  no  divisor  left.  Hence,  64X1^^3  must 
be  the  answer. 

N.  B, — Let  the  pupil  remember  that  this  is  a  gen- 
eral and  universal  method,  equally  applicable  to  any 
per  cent,  or  any  required  time,  and  all  other  rules 
must  be  reconcilable  to  it  ;  and,  in  fact,  all  other 
rules  are  but  modifications  of  this. 

Bankers'  Method  of  Computing  Interest   at 
6  per  cent,  for  any  Number  of  Days. 

Rule. — Draw  a  perpendicular  line,  cutting  off  the 
two  right  hand  figures  of  the  $,  and  you  have  the  inte- 
rest for  60  days  at  6  per  cent. 

Note. — The  figures  on  the  left  of  the  line  arc  dol- 
lars, and  those  on  the  right  are  decimals  of  dollars. 

Example  1. — What  is  the  interest  of  $423,  60  days, 
at  6  per  cent.  ? 

$423=the  principal, 

$4  I  23  cts.=interest  for  60  days. 

Note. — When  the  time  is  more  or  less  than  60 
days  first  get  the  interest  for  60  days,  and  from  that 
to  the  time  required. 

Example  2. — What  is  the  interest  of  $124  for  15 
days,  at  6  per  cent.  ? 

Days.  Days. 

15=1  of  60 
$124=principal. 

4)1  I  24  cts.=interest  for  60  days. 


31  cts.=interest  for  15  days. 


COMilERCIAL   ARITHMETIC.  49 

Example  3.— What  is  the  interest  of  $123.40  for  90 
da^^s,  at  6  per  cent.  ? 

Days.  Days.  Days. 
90=60+30 

$123.40=principal. 

2)1  I  2340=interest  for  60  days. 
6170=interest  for  30  days. 


Ans.  $1  I  851=interest  for  90  days. 

Example  4.— What  is  the  interest  of  8324  for  15 
days,  at  6  per  cent.  ? 

Days.  Days.  Days. 

15=60+15 
|;324=principal. 

4)3  I  24  cts.  interest  fori)0  days. 
81  cts.  interest  for  15  days 


Ans.  $4  I  05  cts.  interest  for  75  days. 

Remark.— This  system  of  computing  interest  is 
very  ea.sy  and  simple,  especially  when  the  days  are 
aliquot  parts  of  60,  and  one  simple  division  will  suf- 
fice. It  is  used  extensively  by  a  large  majority  of 
our  most  prominent  bankers  ;  and,  indeed,  is  taught 
))y  most  all  commercial  colleges  as  the  shortest  sys- 
tem of  computing  interest. 

Method  of  Calculating  at  Different  per  cents. 

This  principle  is  not  confined  alone  to  6  per  cent., 
as  many  suppose  who  teach  and  use  it.  It  is  their 
custom  first  to  find  the  interest  at  6  per  cent.,  and 
from  that  to  other  per  cents  ;  but  it  is  equally  ap- 
plicable for  all  per  cents.,  from  1  to  15,  inclusive. 

The  following  table  shows  the  different  per  cents., 
with  the  time  that  a  given  number  of  $  will  amount 
to  the  same  number  of  cents  when  placed  at  interest: 


50  THE    MERCHANTS    AND    MECHANICS' 

Rule. — Draw  a  perpendicular  line,  cutting  off  the 
two'right  hand  figures  of  $,  and  you  have  the  interest 
at  the  following  per  cents. : 

Interest  at  4  per  cent,  for  90  days. 
Interest  at  5  per  cent,  for  12  days. 
Interest  at  6  per  cent,  for  60  days. 
Interest  at  t  per  cent,  for  52  days. 
Interest  at  8  per  cent,  for  45  days. 
Interest  at  9  per  cent,  for  40  days. 
Interest  at  10  per  cent,  for  36  days. 
Interest  at  12  per  cent,  for  30  days. 
Interest  at  t-30  per  cent,  for  50  days. 
Interest  at  5-20  per  cent,  for  *I0  days. 
Interest  at  10-40  per  cent,  for  35  days. 
Interest  at  7J  per  cent,  for  48  days. 
Interest  at  4J  per  cent,  for  80  days. 

Note. — The  figures  on  the  left  of  the  perpendicu- 
lar line  are  dollars,  and  on  the  right  decimals  of  dol- 
lars.    If  the  dollars  are  less  than  1 0  prefix  a  cipher. 

Example  1. — What  is  the  interest  of  $120  for  15 
days  at  4  per  cent.  ? 

Days.  Days. 

$120=principal.  15=J-  of  90 


6)1 


20  cts.=interest  for  90  days. 
20  cts.=interest  for  15  days. 


Example  2. — What  is  the  interest  of  $132  for  l"^ 
days,  at  t  per  cent.  ? 

Days.  Days. 

$132=principal.  13=J  of  52. 


4)1 


32  cts.=interest  for  52  days. 

33  cts.=interest  for  13  days. 


Example  3. — What  is  the  interest  of  ^520  for  S> 
days,  at  8  per  cent.  ? 


COMMERCIAL    ARITHMETIC.  61 

Days.  Days. 

$520=prmcipal.  9=-^  of  45. 

6)5  I  20  cts.=interest  for  45  days. 
$1  I  04  cts.=interest  for  9  days. 
Example  4.— What  is  the  iuterestof  $462  for  64 
days,  at  7 1  per  cent.  ? 

Days.  Days.  Days, 

$462=prineipal.  61=48+16. 


3)4 
$1 


62  cts.=interest  for  48  days. 
54  cts.=interest  for  16  days. 


$6  I  16  cts.=iiitercst  for  64  days. 

Remark. — We  have  now  illustrated  several  exam- 
ples by  the  different  per  cents.,  and  if  the  student 
will  study  carefully  the  solution  to  the  above  exam- 
ples, he  will  in  a  short  time  be  very  rapid  in  this 
mode  of  computing  interest. 

Note. — The  preceding  mode  of  computing  interest 
is  derived  and  deduced  from  the  cancelling  system, 
as  the  ingenious  student  will  readily  see.  It  is  a 
short  and  easy  way  of  finding  interest  for  days  when 
the  days  are  even  or  aliquot  parts  ;  but  when  they 
are  not  multiples,  and  three  or  four  divisions  are  ne- 
cessary, the  cancelling  system  is  much  more  simple 
and  easy.  We  will  here  illustrate  an  example  to 
show  the  diflference. 

Required,  the  interest  of  $420  for  49  days,  at  6 
per  cent.: 

bankers'  method.  cancelling  method. 

20  cts.=int.  for  60  days. 


2)4 

2)2 
5)1 
3) 


$—$$ 


10  cts.==int.  for  30  days. 
05  ct3.=int.  for  15  days. 

21  cts.=int.  for  3  days.  

7  cts.=int.  for  1  day.         $3,430  Ans. 


^^0—70 

0 

49 
70 


$3  I  43  cts.=int  for  49  days. 


52  THE    MERCHANTS    AND    MECHANICS* 

The  cancelling  method  is  much  more  brief — we 
simply  cancel  6  in  36,  and  the  quotient  6  into  420  ; 
there  is  no  divisor  left ;  hence,  tOX49  gives  the  in- 
terest at  once. 

If  the  time  had  been  15  or  20  days,  the  Bankers' 
method  would  have  been  equally  as  short,  because 
15  and  20  are  aliquot  parts  of  60.  The  superiority 
of  the  cancelling  system  above  all  others  is  this,  it 
takes  advantage  oi  i\\e  principal  as  well  as  the  time. 

For  the  benefit  of  the  student,  and  for  the  conve- 
nience of  business  men,  we  will  investigate  this  sys- 
tem to  its  full  extent,  and  explain  how  to  take 
advantage  of  the  i^nrzc/paZ  when  no  advantage  can 
be  taken  of  the  days.  This  is  one  of  the  most  impor- 
tant characteristics  of  interest,  and  very  often  saves 
much  labor.  It  i^hould  he  used  when  the  days  are  not 
even  or  aliquot  parts. 

The  following  table  shows  the  different  sums  of 
money  (at  the  different  per  cents. )that  bear  one  cent 
interest  a  day  ;  hence,  the  time  in  days  is  always  the 
interest  in  cents  ;  therefore,  to  find  the  interest  on 
any  of  the  following  notes,  at  the  per  cent,  attached 
to  it  in  the  table,  we  have  the  following 

Rule. — Draw  a  perpendicular  line,  cutting  off  the 
two  right  hand  figures  of  the  days  for  cents,  and  you 
have  the  interest  for  the  given  time. 

Interest  of  $00  at  4  per  cent,  for  1  day  is  1  cent. 
Interest  of  $12  at  5  per  cent,  for  1  day  is  1  cent 
Interest  of  $G0  at  6  per  cent,  for  1  day  is  1  cent 
Interest  of  $52  at  t  per  cent,  for  1  day  is  1  cent 
Interest  of  $45  at  8  per  cent,  for  1  day  is  I  cent. 
Interest  of  |40  at  9  per  cent,  for  1  day  is  1  cent. 
Interest  of  $36  at  10  per  cent,  for  1  day  is  1  cent. 
Interest  of  130  at  12  per  cent,  for  1  day  is  1  cent. 
Interest  of  ^50  at  7-30  per  cent,  for  1  day  is  1  ceat. 


COMMERCIAL    ARITHMETIC.  63 

Interest  of  $10  at  5.20  per  cent,  for  1  day  is  1  cent. 
Interest  of  $35  at  10.40  per  cent  for  1  day  is  1  cent. 
Interest  of  $48  at  1|  per  cent,  for  1  day  is  1  cent. 
Interest  of  $80  at  4  J-  per  cent,  for  1  day  is  1  cent. 
Interest  of  $24  at  15  per  cent,  for  1  day  is  1  cent. 

Note. — The  7-30  Government  Bonds  are  calculated 
on  the  base  of  305  days  to  the  year,  and  the  5-20s 
and  10-403  on  the  base  of  364  days  to  the  year. 

Problems  Solved,  by  both  Methods. 

AVe  will  now  solve  some  examples  by  both  meth- 
ods to  further  illustrate  this  system,  and  for  the  pur- 
pose of  teaching  the  pupil  how  to  use  his  judgment. 
He  will  then  have  learned  a  rule  Ttiore  valuable  than 
all  others. 

Example  5. — What  is  the  interest  on  $180  for  75 
days,  at  6  per  cent.  ? 

Operation  by  taking  advantage  of  the  dollar. 

75=the  days.  "  $60X3^$180. 

$0  I  islets  =the  interest  of  $60  for  75  days. 
I     3  Multiply  by  3.    . 

Ans.  $2  I  25  cts.=the  interest  of  $180  for  75  days. 
Operation  by  the  Bankers'  method. 
$180=the  principal.  60da.+15da.=75da. 


4)11 


80  cts.=the  interest  for  60  days. 
45  cts.=the  interest  for  15  days. 


Ans.  $2  I  25  cts.=the  interest  for  75  days. 

By  the  first  method  we  multiplied  by  3,  b<»cftuse 
3X$60=$180.  By  the  second  method  we  addeA  on 
J,  because  60da.-|-«j^da.==75da. 


54 


THE    MERCHANTS    AND    MECHANICS^ 


^  N.  B. — When  advantage  can  be  taken  of  both 
time  and  principal,  if  the  student  wishes  to  prove  his 
work  he  can  first  work  it  by  the  Bankers'  method, 
and  then  by  taking  advantage  of  the  principal,  or 
vice  versa.  And  as  the  two  operations  are  entirely 
different,  if  the  same  result  is  obtained  by  each,  he 
may  fairly  conclude  that  the  work  is  correct. 

LIGHTNING    METHOD    OF   COMPUTING 
INTEREST 

ON     ALL    NOTES    THAT     BEAR     $12     PER     ANNUM,     OR     ANY 
ALIQUOT    PART    OR   MULTIPLE    OF    1 12. 

If  a  note  bears  $12  per  annum  it  will  certainly 
bear  $1  per  month  ;  hence,  the  time  in  months 
would  be  the  interest  in  dollars  and  the  decimal 
parts  of  a  dollar  ;  therefore,  when  the  note  bears 
$12  per  annum  we  have  the  following 

Rule. — Reduce  the  years  to  months,  add  in  the  given 
months,  and x)lace  one  third'of  the  days  to  the  right  of 
this  number,  and  you  have  the  interest  in  dimes. 

Example  1. — Required,  the  interest  of  $200  for  S 
years  T  months  and  12  days,  at  6  per  cent. 

200  ^  of  12  days==4. 

Yr.  Mo.  Da. 


$12.00=int.  for  1  yr.  8    1    12=43.4mo. 

Hence,  43.4  dimes,  or  $43.40  cts.  Ans. 

We  see  by  inspection  that  this  note  bears  $12 
interest  a  year  ;  hence,  the  time  reduced  to  months, 
with  one  third  of  the  days  to  the  right,  is  the  inte- 
rest in  dimes.  If  this  note  bore  $6  a  year  instead 
of  $12  we  would  take  one  half  of  the  above  int^- 


COMMERCIAL   ARITHMETIC.  55 

rest ;  if  it  bore  $18  instead  of  $12,  we  would  add 
one  half;  if  It  bore  $24  instead  of  12  we  would  mul- 
tiply by  2,  etc. 

Example  2. — Required,  the  interest  of  $150  for  2 
years  6  months  and  13  days,  at  8  per  cent. 

150  I  of  13  days=4|. 

8 

Yr.  Da.  Mo. 

$12.00=int.  for  1  jt.  2  5  13=29.41  nios. 

Hence  $29.4J  dimes,  or  $29.43^  cts.  Ans. 

We  see  by  inspection  that  this  note  bears  $12 
interest  a  year  ;  hence,  the  time  reduced  to  months, 
with  one  third  of  the  days  placed  to  the  right,  gives 
the  interest  at  once. 

Example  3. — Required,  the  interest  of  $160  for  11 
years  11  months  and  11  days,  at  *I^  per  cent. 

160  J  of  11  days==8|. 

Tr.    Mo.  Da. 


$12.00=int.  for  1  yr.         11  11  11==143.3§  mos. 
Hence,  $143.3§  dimes,  or  $140.36§  cts.  Ans. 

WHEN  THE  INTEREST  IS  MORE  OR  LESS  THAN  $12  A  YEAR. 

Rule. — First  Jind  the  interest  for  the  given  time  on 
the  base  o/'$12  interest  a  year;  then,  if  the  interest  on, 
the  note  is  only  $6  a  year,  divide  by  2-,  ?/  $24  a  year, 
nuUiply  by  2-,  if^lSa  year,  add  on  one  half,  etc. 

Example  1. — What  is  the  interest  of  $300  for  4 
/'^ars  7  months  and  18  days,  at  6  per  cent.  ? 


50  THE    MERCHANTS    AND    MECHANICS' 

J  of  18  da5-s=6. 
300  4yr.  7mo.  18da.=55.6mo. 

e 


$18.00=-^iKt.  for  1  ye&U         2)55.6,  int.  at  $12  a  year. 
$18=11  times  $12.  278 

$83.4  Ans. 

If  the  interest  was  $12  a  year  $55.60  would  be  the 
answer,  because  55.6  is  the  tim^  reduced  to  months; 
but  it  bears  $18  a  year,  or  1^  times  12  ;  hence,  1^ 
times  55.6  gives  the  interest  at  once. 

Example  2. — required,  the  interest  of  $150  for  3 
years  9  months  and  2T  days,  at  4  per  cent. 

J  of  2T  days=9. 

150  3yr.  9mo.  21  da.=45.9mo. 

4  2)45.9,  int.  at  $12  a  year. 


^6.00=int.  for  1  year.         $22.95   Ans. 
$6=1  times  $12.     . 

If  the  interest  was  $12  a  year  $45.90  wOAild  be  the 
answer — because  245.9  is  the  time  reduced  to  months; 
but  it  bears  $6  a  year,  or  J  times  12  :  hence,  ^  times 
45.9  gives  the  interest  at  once. 

Remark. — We  have  now  fully  explained  our  rsfttliod 
of  computing  interest  at  the  three  diflTerent  bases. 
Any  and  every  problem  in  interest  can  be  solved  by 
one  of  these  three  bases.  Some  problems  can  be 
solved  easier  by  one  base  than  another.  ■M^hero 
the  days  are  divisible  by  3,  and  their  number,  &.n- 
nexed  to  the  months,  divisible  by  12,  it  is  the  short 
est   and   best  method  to  use  the  base  at  one  pe» 


COMMERCIAL    AKITHMETIC.  bl 

cent.  By  using  one  or  the  other  of  these  three  oases 
the  student  can  avoid  the  use  of  vulgar  fractions. 
The  student  must  study  these  three  principles  care- 
fully, and  learn  to  adopt  readily  the  base  best  suited 
to  the  problem  to  be  solved. 

PARTIAL  PAYMENTS  ON  NOTES,  BONDS 
AND  MORTGAGES. 

To  compute  interest  on  notes,  bonds  and  mort- 
gages,  on  which  partial  payments  have  been  made, 
two  or  three  rules  are  given.  The  following  is  called 
the  common  rule,  and  applies  to  cases  where  the 
time  is  short,  and  payments  made  within  a  year  of 
each  other.  This  rule  is  sanctioned  by  custom  and 
common  law;  it  is  true  to  the  principles  of  simple 
interest,  and  requires  no  special  enactment.  The 
other  rules  are  rules  of  law,  made  to  suit  such  cases 
as  require  (either  expressed  or  implied)  annual  in- 
terest to  be  paid,  and,  of  course,  apply  to  no  business 
transactions  closed  within  a  year. 

Rule. — Compute  the  wiereAit  of  the  principal  sum 
for  the  whole  time  to  the  day  of  settlement,  and  find  the 
amount.  Compute  the  interest  on  the  several paijments 
from  the  time  each  was  paid  to  the  daij  of  settlement; 
add  the  several  payments  and  the  interest  on  each  to- 
getlier  and  call  the  sum  the  amount  of  the  payments; 
subtracting  the  amount  of  the  payments  from  the  amount 
of  the  principal  will  leave  the  sum  due. 

Example. — A  gave  his  note  to  B  for  $10,000  ;  at 
the  end  of  4  months  A  paid  $G,000,  and  at  the  expi- 
ration of  another  4  months  he  paid  an  additional 
sum  of  $3,000  ;  how  much  did  he  owe  B  at  the  close 
of  the  year? 


58  THE    MERCHANTS    AND    MECHANICS' 


BY   THE    COMMON    RULE. 

Principal $10,000 

Interest  for  the  whole  time 600 


Amount $10,600 

ist  payment. .  . |6,000 

Interest,  8  months. .       240 

2d  payment 3,000 

Interest,  4  months.  '        60 


Amount $9,300  9,300 


Due $1,800 

PROBLEMS  IN  INTEREST. 

There  are /owr  parts  or  quantities  connected  with 
each  operation  in  interest ;  these  are  the  Principal, 
Bate  per  cent.,  Time,  Interest  or  Amount. 

If  any  three  of  them  are  given  the  other  may  be 
found. 

Principal,  interest  and  time  given  to  find  the  rate 
per  cent. 

Example  1. — At  what  rate  per  cent,  must  $500  be 
put  on  interest  to  gain  $120  in  4  years  ? 

OPERATION.  BY  ANALYSIS. 

$500  The  interest  of  $1 

.01  for  the  given  time  at 

1  per  cent,  is  4  cts. 
$500  will  be  500 
times  as  much=500 
X.04==$20.  Then, 
if  $20  give  1  j 
$120  will  gi\ 
=6  per  cent. 


5.00  $500   will    be     500 

4  times  as  much=500 


20.00)120.00(6  per  cent.  Ans.      if  $20  give  1  per  ct 
120.00  $120  will  give  J^^' 


COMMERCIAL    ARITHMETIC.  59 

Rule. — Divide  the  given  interest  by  tlie  interest  of 
thrr  given  sum  at  one  per  cent,  for  the  given  time,  and 
the  quotient  will  be  the  rate  per  cent,  required. 

Principal,  interest  and  rate  per  cent,  r;iven  to  find 
the  time. 

Example  2. — How  lonj^  must  $500  be  on  interest 
at  6  per  cent,  to  gain  $120  ? 

operation.  analysis. 
$500  We  Cnd  the  inte- 
.06  rest  of$l  at  the  given 
.  rate  for  one  year  is  six 


30.00)120.00(4  years.  Ans.  cents.     $500  wiil  be, 

120.00  therefore,  600   times 

as  mnch  =  500X.06 


=$30.00.     Xow,  if  it  take  1  year  to  gain  $30,  it  will 
require  Jj^ji  to  gain  $120=4  years.    Ans. 

EQUATION  OF  PAYMENTS. 

Equation  of  Payments  is  the  process  of  finding  the 
equalized  or  average  time  for  the  payment  of  several 
sums  due  at  different  times,  without  loss  to  either 
party. 

To  find  the  average  or  mean  time  of  payment  when 
the  several  sums  have  the  same  date. 

Rule. — Multiply  each  payment  by  the  time  that  must 
elapse  before  it  becomes  due ;  then  divide  the  sum  of 
these  products  by  the  sum  of  the  payments,  and  the  quo-, 
tient  ivilt  be  the  average  time  required. 

Note. — When  a  payment  is  to  be  made  down  it 
has  no  product,  but  it  must  be  added  with  the  other 
payments  in  finding  the  average  time. 

Example. — I  purchased  goods  to  the  amount  ot 


CO 


THE    MERCHANTS    AND    MECHANICS' 


5X400--2000 
8X500=4000  ■  I 


Sl,200  ;  ^300  of  whicli  I  am  to  pay  in  4  months, 
$400  in  5  months,  and  500  in  8  months.  IIow  long 
a  credit  ought  I  to  receive  if  I  pay  the  whole  sum  at 
once  ?     Ans.  6  months. 

!Mo.  Mo.  (     A  credit  on  $300  for  4  months  is  the 

^\/oAA___i  9Arj  Jsame    as  tlie    credit   on    $1  for    1200 

A  credit  on  $400  for  5  months  is  the 
[same  as  the  credit  on  ^i  fur  2000 
months. 

A  credit  on  $500  for  6  mrnths  is  the 

I  same    as   the    credit    on   tl  for  4000 

(months. 

_  ^_^^.  .         ^  ,  Therefore,  I  should  have  the  same 

1200)7200(6  mo.  credit  as  a  credit  on  n  for  7200 mos. ; 

*7  0An  ^^^    ^"    ?1200,   the    whole    sum,   one 

'-^"^  twelve  hundredth  part  of  7200  montlis, 

which  is  6  mouths. 

This  rule  is  the  one  usually  adopted  by  merchants, 
although  not  strictly  correct ;  still,  it  is  sufficiently 
accurate  for  all  practical  purposes 

To  lind  the  average  or  mean  time  of  payment 
when  the  several  sums  have  different  dates. 

Example. — Purchased  of  James  Brown,  at  sundry 
times  and  on  various  terms  of  credit,  as  by  the  state- 
ment annexed.  When  is  the  medium  time  of  pa}^- 
ment  ? 

Jan.  ],  a  bill  am'ting  to  $360,  on  3  months'  credit. 
Jan.    15,   do.         do.  186,  on  4  months'  credit. 

March  1,   do.         do.  450,  "on  4  months'  credit. 

May    15,- do.         do.  300,  on  3  months' credit. 

June   20,   do.         do.  500,  on  5  months'  credit. 

Ans.  July  24,  or  in  115  d%.ys. 
Due,  April    1,  $360. 

"      Mav    15,     186X  44=     8184 

''      Jufy     1,     450X  91=  40950 

"      Aug.  15,     300X136=  40800 

"      Kov.  20,     500X233=116500 


1196-f-into  )206434(114||f  days. 


COMMERCIA!.    ARITHMETIC.  Gl 

We  first  find  the  time  when  each  of  the  bills  will 
become  due.  Then,  since  it  will  shorten  the  opera- 
tion and  bring  the  same  result,  we  take  the  time  lohen 
the  first  bill  becomes  due,  instead  of  its  date,  for  the 
period  from  which  to  compute  the  average  time. 
Now,  since  April  1  is  the  period  from  which  the 
laverage  time  is  computed,  no  time  will  be  reckoned 
on  the  first  bill,  but  the  time  for  the  payment  of  the 
second  bill  extends  44  days  beyond  April  1,  and  we 
multiply  it  by  44. 

Proceeding  in  the  same  manner  with  the  remain- 
ing bills,  we  find  the  average  time  of  payment  to  be 
114  days  and  a  fraction  from  April  1,  or  on  the  24th 
of  July. 

Rule. — Find  the  time  when  each  of  the  sums  be- 
comes due,  and  midliply  each  sum  by  the  number  of 
days  from  the  time  of  the  earliest  payment  to  the 
payment  of  each  sum  respectively;  then  jjroceed  as 
in  the  last  rule,  and  the  quotient  'will  be  the  average 
time  required,  in  days,  from  the  earliest  2^ayment. 

Note. — Nearly  the  same  result  may  be  obtained 
by  reckoning  the  time  in  months. 

In  mercantile  transactions  it  is  customary  to  give 
a  credit  of  from  3  to  9  months  on  bills  of  sale.  Mer- 
chants, in  settling  such  accounts  as  consist  of  vari- 
ous items  of  debit  and  credit  for  difi'erent  times, 
generally  employ  the  following 

Ilui-E. — Place  on  the  debtor  or  credit  side  such  a  sum 
{which  may  be  called  merchandise  balance)  as  will  bal- 
ance the  account.  Multiply  the  number  of  dollars  in 
each  entry  by  the  number,  of  days  from  the  time  the 
entry  was  m,ade  to  the  time  of  settlement,  and  the  mer- 
chandise balance  by  the  number  of  days  for  which  credit 


62  THE  MERCHANTS  AND  MECHANICS* 

was  given.  Then  multiply  the  difference  between  the 
sum  of  the  debit  and  the  sum  of  the  credit  products  by 
the  interest  of  ^1  for  one  day;  this  product  will  be  the 

INTEREST  BALANCE. 

When  the  sum  of  the  debit  products  exceeds  the  sum 
of  the  credit  products  the  interest  balance  is  in  favor  of 
the  debit  side;  but  when  the  sum  of  the  credit  products 
exceeds  the  sum  of  the  debit  products  it  is  in  favor  of 
the  credit  side.  Now,  to  the  merchandise  balance  add 
the  interest  balance,  or  subtract  it,  as  the  case  may  re- 
quire, and  you  obtain  the  cash  balance. 

A  lias  with  B  the  following  account: 


1849. 

Br.     1849. 

Cr. 

Jan.  2.    To  merchandise, 

$200    Feb.  29. 

By  merchandise, 

$100 

April  20.  "                «' 

400    May  10. 

"               .. 

300 

If  interest  is  estimated  at  T  per  cent.,  and  a  credit 
of  60  days  is  allowed  on  the  different  sums,  what  is 
the  cash  balance  August  20,  1849  ?     Ans.  $206.54. 

Explanation. — Without  interest  the  cash  balance 
Avould  be  1200. 

The  object  of  these  changes  is  to  give  the  learner 
an  accurate  and  complete  knowledge  of  numbers 
and  of  division,  and  the  result  is  not  the  only  ob- 
ject sought  for,  as  many  young  learners  suppose. 

How  many  times  is  t5  contained  in  5*15  ?  or  divide 
5t5  by  15.     Ans.  t|. 

Divide  800  by  121.     Quotient,  64. 

Divide  2t  by  16|.     Quotient,  \^^^,  or  Iff 

A  person  spent  $6  for  oranges,  at  6J  cents  apiece, 
how  many  did  he  purchase  ?     Ans.  96. 

When  two  or  more  numbers  are  to  be  multiplied 
together,   and  one  or  more  of  them  have  a  cipher 


COMMERCIAL    ARITHMETIC.  63 

^m  tAe  nglib,  as  24  by  20,  we  may  take  the  cipher 
from  one  number  and  annex  it  to  the  other  without 
affecting  the  product :  thus  24X20  is  the  same  as 
240X2  ;  286X1300=28600X13  ;  and  350X^0X40 
=35X^X^^X1000,  efc. 

Every  fact  of  this  hind,  though  extremely  simple,  unll 
be  very  useful  to  those  ivho  wish  to  be  skilful  in  opera- 
tion. 

Note.— If  there  are  ciphers  at  the  right  hand  either 
of  the  multiplier  or  multiplicand,  or  of  both,  they 
may  be  neglected  to  the  close  of  the  operation,  when 
they  must  be  annexed  to  the  product. 

Remark. — ^We  now  give  a  few  examples  for  the 
purpose  of  teaching  the  pupil  how  to  use  his  judg- 
ment ;  he  will  then  have  learned  a  rule  more  valua- 
ble than  all  others. 

Multiplication  and  Division  Combined. 

When  it  becomes  necessary  to  multiply  two  or 
more  numbers  together,  and  divide  by  a  third,  or  by 
a  product  of  a  third  and  fourth,  it  must  be  literally 
done  if  the  numbers  are  prime.     For  example  : 

Multiply  19  by  13  and  divide  that  product  by  T. 

This  must  be  done  at  full  length,  because  the  num- 
bers are  prime,  and  in  all  such  cases  there  will  result 
Skfraction. 

But  in  actual  business  the  problems  are  almost  all 
reducible  by  short  operations,  as  the  prices  of  arti- 
cles or  amount  called  for  always  corresponds  with 
Bome  aliquot  part  of  our  scale  of  computation  ;  and 
when  two  or  more  of  the  numbers  are  co?7iposi^e  num- 
bers the  work  can  always  be  contracted. 

Example. — Multiply  375  by  7,  and  divide  that  pro- 
duct by  21.     To  obtain  the  answer  it  is  sufficient  to 


64 


THE    MERCHANTS    AND    MECHANICS^ 


divide  375  by  3,  which  gives  125.    The  1  divides  the 
21,  and  the  factor  3    remains  for  a  divisor. 

Here  it  becomes  necessary  to  lay  down  a  j^lan  of 
opei^aiion:  Draw  a  perpendicular  line,  and  place  all 
numbers  that  are  to  be  multiplied  together  under 
each  other  on  the  right  hand  side,  and  all  numbers 
that  are  divisors  under  each  other  on  the  left  hand 
side. 

Examples. 

Multiply  140  by  36  and  divide  that  product  by  84. 
We  place  the  numbers  thus  : 


84 


140 
36 


We  may  cast  out  equal  factors  from  each  side  of 
the  line  without  affecting  the  result.  In  this  case  12 
will  divide  84  and  36,  then  the  numbers  will  stand 
thus  : 

140 
3 

But  *I  divides  140  and  gives  20,  which,  multiplied 
by  3,  gives  00  for  the  result. 

Multiply  4783  bv  39,  and  divide  that  product 
by  13. 

4783 

n  3 

Three  times  4783  must  be  the  result. 


X^ 


Multiply  80  by  9,  that  product  by  21,  and  divide 
the  whole  by  the  product  of  OOX^XU. 


00 
6 

X^ 


U  4 
9 

n  3 


COMMERCIAL    ARITHMETIC.  65 

In  Ihe  foregoing  divide  60  and  80  by  20,  and  14 

and  21  by  7,  and  tiiose  numbers  will  stand  canceUjd, 
with  3  and  4,  2  and  3  at  their  sides. 

Now,  the  product  3X^X2  on  the  divisor  side  is 
equal  to  4  times  9  on  the  other,  and  the  remaining 
3  is  the  result. 

General  Rules        Cancellation. 

1.  Draw  a  perpendicular  line  ;  observe  this  line 
represents  the  sign  of  equality.  On  the  right  hand 
side  of  this  line  place  dividends  only,  on  the  left 
hand  £ide  place  divisors  only.  Having  placed  divi- 
dends on  the  right  and  divisors  on  the  left,  as  above 
directed, 

2.  Notice  whether  there  are  ciphers  both  on  the 
right  and  left  of  the  line  ;  if  so,  erase  an  equal  num- 
ber from  each  side. 

3.  Notice  whether  the  same  number  stands  both 
on  the  right  and  left  of  the  line  ;  if  so,  erase  them 
both. 

4.  Notice  again  if  any  number  Ofl  either  side  of 
the  line  will  divide  any  number  on  the  opposite  side 
without  a  remainder  ;  if  so,  divide  and  erase  the  two 
numbers,  retaining  the  quotient  fi^^ure  on  the  side  of 
the  larger  number. 

5.  See  if  any  two  numbers,  one  on  each  side,  can 
be  divided  by  any  assumed  number  without  a  re- 
mainder ;  if  so,  divide  them  by  that  number  and 
retain  only  their  quotients.  Proceed  in  the  same 
manner  as  far  as  practicable,  then 

6.  Multiply  all  the  numbers  remaining  on  the  right 
hand  side  of  the  line  for  a  dividend,  and  those  re- 
maining on  the  left  for  a  divisor. 

1.  Divide,  and  the  quotient  is  the  answer. 


pg  THE   MERCHANTS    AXP    MECHANICS* 


THE    MILLER'S    RULE     FOR    WEIGHING 
WH.KA.T. 

"Wlieat  weighing  58  pounds  and  upwards  por 
bushol  is  considered  merchantable  wheat,  and  60 
poun-is  of  merchantable  wheat  make  a  standard 
bushel.  Hence,  wheat  weighing  less  than  60  pounds 
per  bushel  will  lose  in  making  up  ;  but,  weighing 
more,  it  will  gain. 

When  wheat  weighs  less  than  58  pounds  per 
bushel  it  is  customary,  on  account  of  the  inferior 
yield  of  li^ht  wheat,  to  take  two  pounds  for  one  in 
making  up  the  weight;  hence,  it  will  take  63  pounds 
to  make  up  a  bushel,  provided  the  wheat  weighs  but 
67,  and  64  if  the  wheat  weighs  but  56  pounds  per 
bushel. 

Case  I. — Tv>  change  merchantable  wheat  to  stan- 
dard weight. 

Rule, — Brin^ij  the  whole  quantity  of  wheat  to 
pounds  and  divide  by  60. 

Example  1. — Eow  many  standard  bushels  of  wheat 
are  in  150  bushsl^,  each  weighing  58  pounds  ? 


150 

58 

Or,  each  bushel  lacks  2  lbs.;  150 

2 

1200 
750 

,870,0 

6,0)30,0 

From  150  bush.     Deficiency,   5 
Take       5                                  

Ans.  145  bush.      1  «aves  145,  the  answer. 


COMMERCIAL    ARITHMETIC.  67 

2.  How  many  standard  bushels  of  wheat  are  in  80 
bushels  45  pounds,  weighing  63  ? 

Bush.  lbs. 
80      45  Or,  80  bus. 

63  3 


285              6,0)24,0  excess  of  w'gt. 
480  


4  bus. 


6,0)508,5  80   45  lbs. 

Ans.  84  bu.  46  lbs.=84|  bu.  Ans.  84  bu.  45  lbs.  or  3  p. 

3.  How  many  standard  bushels  of  wheat  are  in 
175  bushel?  37  pounds,  weighing  59  ?  Ans.  172  bus. 
42  lbs. 

4.  How  many  standard  bushels  are  in  100  bushels 
15  pounds,  weighing  62  pounds  per  bushel?  Ans. 
103  bus.  35  lbs. 

Case  II. — When  wheat  weighs  less  than  58. 

Rule. — Bring  the  whole  quantity  to  pounds,  and  di- 
vide by  as  many  pounds  as  make  a  standard  bushel  of 
such  wheat. 

Example  1. — How  many  bushels  of  good  wheat 
are  equal  to  100  bushels  weighing  57  ? 

100  Or,  6  lbs.  per  bus.=600  lbs. 

57  03)600(9  bus.  33  lbs.  defect. 

567 


63)5700(90  bus.  30  lbs. 


567  From  100  bus.    33 

—  Take      9  33 

30  lbs.  

Ans.  90  30 


68  THE    MERCHANTS    AND    MECHANICS^ 

Note. — The  odd  pounds  in  the  above  and  follow- 
ing results  are  also  subject  to  a  small  drawback, 
viz.,  1  lb.  in  every  21  when  the  wheat  weighs  6t  ;  1 
in  16  when  it  weighS'  56,  and  so  on  ;  consequently 
the  above  ought,  in  strictness,  to  be  90  bushels,  and 
father  more  than  28 i  pounds,  but  millers  seldom 
make  this  deduction. 

2.  How  many  standard  bushels  of  merchantable 
wheat  will  be  equal  to  250  bushels  18  lbs.  weighing 
56  lbs.  per  bushel  ?     Ans.  219  bus.  2  lbs. 

3.  How  much  good  wheat  is  equal  to  1000  bushels 
weighing  55  ?     Ans.  846  bus.  10  lbs. 

Note. — Before  dismissing  this  rule  it  appears  pro- 
per that  a  few  remarks  should  be  made,  in  order  to 
show  the  young  farmer  the  importance  of  under- 
standing it  properly.  There  are  different  methods  of 
"  making  up  wheat"  (i.  e.,  finding  its  merchantable 
value),  and  these  methods  give  different  results  ; 
hence  the  necessity  of  the  subject  being  understood 
by  all  concerned.  I  shall  not  undertake  to  deter- 
mine between  the  farmer  and  the  miller  which  is  or 
which  is  not  the  fair  way;  but,  after  explaining  the 
principle,  leave  them  to  make  liJieir  bargains  as  they 
may  choose. 

If  I  have  a  bushel  of  wheat  that  weighs  but  57 
pounds,  then  six  pounds  of  the  same  kind  a/wheat  will 
be  necessary  to  make  this  a  merchantable  bushel, 
so  that  63  pounds  of  this  quality  of  wheat  will  make 
a  standard  bushel;  and  it  is  upon  this  supposition 
that  the  preceding  calculations  are  founded.  But  a 
number  of  millers  use  a  method  of  calculation  by 
which  they  take  good  wheat  for  the  odd  pounds — i. 
e.,  they  take  a  bushel  full,  say  57  pounds,  of  the 
wheat  they  are  measuring,  and  instead  of  takii»g  six 


COJOIERCIAL    ARITHMETIC.  69 

pounds  more  of  the  same  kind  to  make  ifc  up,  they 
take  six  pounds  of  good  or  merchantable  wheat.  Their 
method  of  calculation  is  as  follows: 

Required,  the  good  wheat  in  1000  bushels  weigh- 
ing 55  ?  Defect,  10  lbs.  per  bus.  1000 

10 

From  1000  bus.  

Take     166  40  6,0)1000,0 


Ans.  833  .20  Bush.  166  40 

We  see  that  this  gives  a  result  nearly  13  bushels 
more  in  the  miller's  favor  than  the  former  method; 
and  this  I  know  to  be  practiced  by  many. 

There  is  another  method  sometimes  used  by  those 
who  arc  not  very  scrupulous  in  their  distinctions  be- 
tween right  and  wrong.  They  find  the  whole  defect 
in  pounds,  and  divide  by  the  weight  of  a  bushel  of 
the  wheat  to  find  how  many  bushels  of  that  kind  of 
wheat  will  make  up  the  defect,  thus: 

Required,  the  good  wheat  in  1000  bushels  weigh- 
ing 55  pounds  per  bushel  ? 

Defect,  10  lbs.  per  bushel=10000  in  all. 


From  1000  bus. 
Take     181  45 

55)10000(181  bus.  45  lbs. 
55 

Ans.  818  10 

450 
440 

100 
55 

45  lbs. 
We  tee  that  this  method  gives  15  bushels  more  to 


TO  THE    MERCHANTS    ANB    MECHANICS* 

the  miller  than  the  last  and  28  more  than  the  first. 
It,  however,  shows  how  much  of  the  same  kind  of 
wheat  must  be  added  to  the  1000  to  make  1000 
bushels  of  good  wheat,  viz.,  181  bushels  45  pounds — - 
for  1181  bushels  45  pounds,  weighing  55,  will  just 
make  1000  bushels  "  made  up  weight."  This  method 
would  be  as  erroneous  as  calculating  discount  by 
the  rule  for  interest. 

Another  method  is  to  take  two  pounds  for  one  up 
to  58,  and  pound  for  pound  afterwards.  To  do  this 
bring  the  whole  quantity  to  pounds,  and  if  the  wheat 
weigh  57  divide  by  61;  if  56,  by  62,  and  so  on.  This 
appears  more  reasonable  than  the  others,  as  it  makes 
less  difference  between  wheat  barely  merchantable 
and  that  which  is  not  quite  so.  By  the  former  rule, 
if  the  wheat  weigh  58,  two  pounds  per  bushel  will 
make  it  up,  but  if  57,  six  pounds  are  necessary;  by 
this  rule  only  one  extra  pound  would  be  taken.  In 
subtracting  the  odd  pounds,  the  lower  number  being 
greatest  (suppose  the  wheat  to  weigh,  say  57),  the 
calculator  may  be  at  a  loss  whether  to  take  from  57, 
60  or  63.  In  this  case,  let  the  running  weight  be 
what  it  may,  we  should  take  from  the  weight  made 
up,  as  in  example  1,  case  2,  we  took  from  63. 

This  subject  is  of  importance  to  both  farmers  and 
millers,  and  if  they  do  riot  attend  to  it  they  deserve 
to  be  cheated. 

SQUARE  AND  CUBE  ROOTS. 

A  number  multiplied  by  itself  once  will  produce 
the  square  of  that  number. 

A  number  multiplied  by  itself  twice  will  produce 
the  cube  of  that  number. 

Extracting  the  root  of  a  number  i»  finding  what 
number  multiplied  into  itself,  once  f^r  square  and 
twice  for  cube,  will  produce  the  giveix  ^Aumber. . 


COMMERCIAL    ARITHMETIC. 


71 


Numbers  possess  the  following  properties: 

1.  A  square  multiplied  by  a  square  will  produce 
a  square. 

2.  A  square  divided  by  a  square  the  quotient  will 
be  a  square. 

3.  A  cube  multiplied  by  a  cube  will  produce  a 
cube. 

4.  A  cube  divided  by  a  cube  the  quotient  will  be 
a  cube. 

5.  If  the  unit  figure  of  a  square  number  is  5  we 
may  multiply  by  the  square  number  4,  and  we  shall 
produce  a  square  whose  unit  period  will  be  ciphers. 

6.  If  the  unit  figure  of  a  cube  is  5  we  may  multi- 
ply by  the  cube  number  8,  and  produce  a  cube 
whose  unit  period  will  be  ciphers. 

The  following  table  should  be  committed  to  mem- 
ory, in  order  to  be  able  to  extract  roots  immediately 
by  inspection: 


Numbers. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Squares. 

1 

4 

9 

16 

25 

36 

49 

64 

81 

100 

Cubes. 

I 

8 

27 

64 

125 

216 

343 

729 

1000 

To  Extract  Square  Root  by  Inspection. 

Commencing  at  the  right  hand,  point  the  number 
oflf  into  periods  of  two  figures  each,  and  the  root  will 
contain  as  many  figures  as  there  are  periods.  From 
the  table  find  the  greatest  square  of  each  period,  and 
these  numbers,  set  down  in  their  order,  will  be  the 
answer  required. 


72  THE    MERCHANTS    AND    MECHANICS^ 

Example  1. — What  is  the  square  root  of  225  ?    Ans» 

15. 

Pointing  it  off  (2.25)  we  find  it  consists  of  two  pe- 
riods, 2  and  25.  By  referring  to  the  table  we  find 
lliat  the  greatest  square  in  2  is  1,  and  the  square  in 
25  is  5=15. 

2.  What  is  the  square  root  of  2025  ?     Ans.  45.       1 
The  greatest  square  in  20  is  4,  and  5  is  the  only- 

number  whose  square  is  5  in  the  unit's  place. 

3.  Extract  the  square  root  of  6561.     Ans.  81. 

As  the  unit  figure  is  1,  and  in  the  table  we  find  1 
only  at  1  and  8i,  we  will  divide  6561  by  81,  and  we 
find  the  quotient  81.  81  is,  therefore,  the  square 
root. 

4.  Extract  the  square  root  of  106729.     Ans.  327. 

As  the  unit  figure  is  9,  if  the  number  is  a  square 
it  must  either  divide  by  9  or  49.  Dividing  by  9  we 
obtain  11881,  which  is  a  prime  number=109.  This 
number  multiplied  by  3,  the  root  of  9,  gives  327. 

5.  Extract  the  square  root  of  451584.     Ans.  672. 

As  the  unit  figure  is  4,  and  in  the  table  of  squares 
we  find  4  only  at  4  and  64,  the  number,  if  a  square, 
must  divide  by  4  or  64,  or  both.  Dividing  by  4,  we 
have  the  factors  4  and  112896.  This  last  factor  ends 
in  6;  therefore,  by  tlie  table,  we  see  it  must  divide 
by  16  or  36.  Dividing  by  36  we  have  the  factors  36 
and  3136.  This  last  factor,  ending  in  6,  we  divide 
by  16,  and  have  the  factors  16  and  196.  Dividing 
this  last  factor  by  4  we  liave  4  and  49.  Take  our 
divisors  and  last  factor,  49,  and  we  have  for  the  ori- 
crinal  number,  4X36X16X^X^9,  the  roots  of  which 
are  2X6X4X2X^=672. 


COMMERCIAL    ARITHMETIC.  73 

6    Extract  the  square  root  of  1225.     Ans.  35. 

Divide  by  the  square  number  25  and  we  have  the 
two  factors  25  and  49  as  equivalent  to  the  given 
number.  Roots  of  these  factors,  5X^=35.  Or,  as 
tho  number  ends  in  25,  we  may  multiply  by  4  and 
we  have  4900,  the  root  of  which  is  70 — which,  divi- 
ded by  2,  the  square  root  of  4,  gives  35. 

Problem. — The  top  of  a  castle  is  45  yards  high 
and  the  castle  is  surrounded  by  a  ditch  60  jards 
wide.  Required,  the  length  of  a  rope  that  will  reach 
from  tho  outside  of  the  ditch  to  the  top  of  the  castle. 
Ans.  75  yards. 

The  usual  rule  for  this  is  to  add  the  square  of  the 
two  sides  and  extract  the  square  root  of  the  sum; 
but  so  nmch  labor  is  never  necessary  when  the  two 
numbers  have  a  common  divisor,  or  when  the  answer 
is  a  composite  number. 

Take  45  and  60  and  divide  them  both  by  15  and 
they  will  be  reduced  to  3  and  4;  square  these,  and 
their  sum  9-|~16=25;  extract  the  square  root  of  25, 
which  is  5,  multiplied  by  15  gives  75, 

When  it  is  requisite  to  multiply  several  numbers 
together  and  extract  the  root  of  their  product,  try  to 
change  them  into  factors  and  extract  the  root  bejfore 
muUiplicaiion. 

To  Extract  Cube  Root  by  Inspection. 

We  can  extract  the  root  of  cube  numbers  by  in- 
spection  when  they  do  not  contain  more  than  two 
periods  of  three  figures  each. 

By  examining  the  table  it  will  appear  evident  that 
if  the  unit  figure  of  the  power  be  1  the  unit  figure  in 
tho  root  will  be  1;  if  it  be  8  the  root  will  be  2;  if 
7  it  will  be  3;  if  4  the  root  will  be  4;  if  5  it  will  bo 


u 


THE    MERCHANTS    AXD    MECHANICS' 


5;  if  6  it  willbe  6;  if  3  it  will  be  t;  if  2  it  will  be  8; 
and  if  it  be  9  the  root  will  be  9. 

Example  1. — Find  the  cube  root  of  111649. 

This  number  consists  of  two  periods.  Compare 
the  ten's  period  with  the  cubes  in  the  table  and  we 
find  that  lit  lies  between  64  and  125.  The  cube 
root  of  the  tens  then  must  be  4.  The  unit  figure  of 
the  unit  period  being  9  the  root  must  be  9;  there- 
fore 49  is  the  root  required. 

2.  What  is  the  cube  root  of  389011  ?     Ans.  13. 

By  looking  in  the  table  we  find  that  the  ten's  pe- 
riod, 389,  lies  between  343  and  512;  the  root  must, 
therefore,  be  1.  The  unit  figure  of  the  unit  period 
being  1  the  root  must  be  3,  therefore  13  is  the  re- 
quired root. 

When  a  cube  has  more  than  two  periods  it  can 
generally  be  reduced  to  two  by  dividing  by  some 
cube  number,  unless  the  root  is  a  prime  number. 

3.  What  is  the  cube  root  of  3241192.     Ans.  148. 
As  the  unit  figure  of  the  unit  period  is  2  the  root 

of  that  period  must  be  8.  Now,  dividing  3241192 
by  8,  we  have  405224,  a  number  consisting  of  but 
two  periods,  the  root  of  which  we  find  by  inspection 
to  be  14 — which,  multiplied  by  2,  the  root  of  8,  gives 
148,  the  root  required. 

4.  What  is  the  cube  root  of  12911815  ?    Ans.  235. 
As  this  number  ends  in  5  we  will  multiply  it  by 

8,  and  if  it  is  a  cube  the  unit  period  will  be  ciphers: 
12911815X8=103623000.  This  number  may  be  re- 
garded as  two  periods — the  unit  period  being  0.  By 
inspection  the  root  of  103623  is  41;  annexing  tlio 


COjrMERCIAL    ARITHMETIC.  75 

cipher  equals  4T0 — which,  divided  by  2,  the  root  of 
8,  gives  235,  the  answer. 

To  find  the  cube  root  of  surds,  very  nearly: 

Rule, — Take  the  nearest  cube  to  the  given  number 
and  call  it  the  assumed  cube.  Double  the  assumed  cube 
and  add  the  number  to  it;  also,  double  the  number  and 
add  the  assumed  cube  to  it.  Take  the  difference  of 
these  sums,  then  say,  as  double  the  assumed  cube  added 
to  the  number  is  to  this  difference,  so  is  the  assumed 
root  to  the  correction.  This  correction,  added  to  or 
subtracted  from  the  assumed  root,  as  the  case  may  re- 
quire, will  give  the  cube  root,  very  nearly. 

5.  What  is  the  cube  root  of  214  ? 

The  nearest  cube  to  214  is  216,  the  root  of  which 
is  6,  and  it  is  evident  the  root  of  214  will  be  less,  and, 
to  state  the  proportion  according  to  the  rule,  we  have 

216X2=432  214X2=428 

214  216 

646  644=2  difference. 

As  646  :  2  ::   6  :  to  correction — 

2X6=12.         646)12.00000(0.0185t+correction. 
646 
5540 
5168 


3720 
3230 


4900 
4522 

As  the  assumed  root  is  more,  we  subtract  the  cor- 
rection : 

6.00000 

0.01857-1- 

5.98143-f-the  required  root,  very  nearly. 


76  THfl    MERCHANTS    AND    MECHANICS' 

A   TABLE   FOR   MEASURING  TIMBER. 


Quartee 
Gjkt. 

Area. 

Quarter 

GlKT. 

Area. 

Quarter 
Girt. 

Area. 

Inches. 

6 

It 

Feet. 

.250 

.272 
.294 
.317 

Inches. 

12 
12| 

Feet. 

1.000 
1.042 
1.085 
1.129 

Inches. 

18 
181 
19 
19i 

Feet. 
2.250 

2.376 
2.506 
2.640 

1 

It 
k 

.340 
.364 
.390 
.417 

13 

13i 
13| 

1.174 
1.219 
1.265 
1.313 

20 
201 
21 
21i 

2.777 
2.917 
3.062 
3.209 

8 

8| 
8| 

.444 
.472 

.501 
.531 

14 
^14i 

14| 

1.361 
1.410 
1.460 
1.511 

22 

221 

23 

231 

3.362  ' 
3.516 
3:673 
3.835 

9 
H 

H 

9J 

.562 
.594 
.626 
.659 

15 

151 
15j 

1.562 
1.615 
1.668 
1.722 

24 
241 
25 
25| 

4.000 
4.168 
4.340 
4.516 

10 

lOi 
lOi 
10| 

,694 
.730 
.766 
.803 

16 

16i 
16| 
16i 

1.777 
1.833 
1.890 
1.948 

I 

26 

261 

27 

271 

4.694 
4.876 
5.062 
5.252 

11 

Hi 

111 

Hi 

.840 
.878 
.918 
.959 

17 

in 

2.006 
2.066 
2.126 
2.1$6 

28 

281 

29 

291 

30 

5.444 
5.640 
5.840 
0.044 
6.250 

COIIMERCIAT.    AKITHMETIC. 


n 


Rule. — (By  the  Carpenters'  Rule.) — Measure  the 
circumference  of  the  piece  of  timber  in  the  middle 
and  take  a  quarter  of  it  in  inches;  call  this  the  ^  girt. 
Then  set  12  on  d  to  the  length  in  feet  on  c,  and  against 
the  girt  in  inches  on  d  you  will  find  the  content  infect 
on  c. 

Example. — If  a  piece  of  round  timber  be  18  feet 
long-,  and  the  quarter  girt  24  inches,  how  many  feet 
of  timber  are  contained  therein  ? 


24  quarter 
24 

girt. 

96 

48 

BY   THK   table. 

596  square. 
18 

Against  24  stands 
Length, 

4.00 

18 

4008 

Product, 

72.00 

576 

Ans.  72  feet. 

144)10368(12  feet. 
1008 

288 
288 

Bg  the  Carpenters'  Rule 
D  :  72  on  c. 

— 12  on  D  :  18  on  c  : 

24  Oil 

Problem  I. — To  find  the  solid  contents  of  squared 
or  four-sided  timber  by  the  Carpenters'  Rule — as  12 
on  D  :  length  ou  c  :  quarter  girt  on  d  :  solidity  on  c. 

Rule  I. — Multiply  the  breadth  in  the  middle  by  the 
depth  in  the  middle,  and  that  product  by  the  lengtlifo^ 
the  solidity. 


78  THE    MERCHANTS    AXD    MECHANICS^ 

Note. — If  the  tree  taper  regularly  from  one  end  to 
the  other  half  the  sum  of  the  breadths  of  the  two 
ends  will  be  the  breadth  in  the  middle,  and  half  the 
sum  of  the  depths  of  the  two  ends  will  be  the  depth 
in  the  middle. 

KuLE  II. — Multiply  the  sum  of  the  breadths  of  the 
two  ends  by  the  sum  of  the  depths,  to  which  add  the 
product  of  the  breadth  and  depth  of  each  end;  one  sixth 
of  this  sum  multiplied  by  the  length  will  give  the  cor- 
rect solidity  of  any  piece  of  squared  timber  tapering 
regularly. 

Problem  II. — To  find  how  much  in  length  will 
make  a  solid  foot,  or  an}'-  other  assigned  quantity  of 
squared  timber,  of  equal  dimensions  from  end  to  end. 

Rule. — Divide  1*128,  the  solid  inches  in  afoot,  or 
the  solidity  to  be  cut  off,  by  the  area  of  the  end  in 
inches,  and  the  quotient  will  be  the  length  in  inches. 

Note. — To  answer  the  purpose  of  the  above  rule 
some  carpenters'  rules  have  a  little  table  upon  them 
in  the  following  form,  called  a  table  of  timber  measure: 


0 

0 

0 

0 

9 

0 

111 

3 

9 

1  inches. 

144 

36] 

16 

9 

5 

4 

2 

2 

1 

1  feet. 

1 

2 

3 

4 

1    5 

6 

1    7 

8 

9 

1  side  of  the  square. 

This  table  shows  that  if  the  side  of  the  square  be 
one  inch  the  length  must  be  144  feet;  if  two  inches  be 
the  side  of  the  square  the  length  must  be  36  feet 
to  make  a  solid  fcot. 

Problem  III. — To  find  the  solidity  of  round  or  un- 
squared  timber. 

Bule  I. — Gird  the  timber  round  the  middle  vnth  a 


COMMERCIAL    ARITHMETIC.  79 

string  ;  one  fourth  part  of  this  girt  squared  and  multi- 
plied by  the  length  will  give  the  solidity. 

Note. — If  the  circumference  be  taken  in  inches 
and  the  length  in  feet,  divide  the  last  product  by 
144. 

Rule  II. — (By  the  Table). — Ifultiply  the  area  cor- 
respondiiig  to  the  quarter  girt  in  inches  by  the  length 
of  the  piece  of  timber  in  feet,  and  the  pyroduct  will  be 
the  solidity. 

Note. — If  the  quarter  girt  exceed  the  table  take 
half  of  it,  and  four  times  the  content  thus  formed 
will  be  the  answer. 

Sow  do  you  do  when  the  timber  tapers  f 

Gird  the  timber  at  as  many  points  as  may  be  ne- 
cessary, and  divide  the  sum  of  the  girts  by  their 
number  for  the  mean  girt,  of  which  take  one  fifth 
and  proceed  as  before. 

If  a  tree,  girting  14  feet  at  the  thicker  end  and  2 
feet  at  the  smaller  end,  be  24  feet  in  length,  how 
many  solid  feet  will  it  contain  ?    Ans.  122.88. 

A  tree  girts  at  five  different  places  as  follows:  In 
the  first  9.43  feet,  in  the  second  7.92  feet,  in  the 
third  6.15  feet,  in  the  4th  4.74  feet,  and  in  the  fifth 
3.16  feet;  now,  if  the  length  of  the  tree  be  17.25  feet, 
what  is  its  solidity  ?     Ans.  54.42499  cubic  feet. 

OF  logs  for  sawing. 

What  18  often  necessary  for  lumber  merchants  f 

It  is  often  necessary  for  lumber  merchants  to  as- 
certain the  number  of  feet  of  boards  which  can  be 
cut  from  a  given  log  ;  or,  in  other  words,  to  find 


80  thl:  merchants  and  jilichanics" 

how  many  logs  will  be  necessary  to  make  a  given 
amount  of  boards. 

WJiat  is  a  standard  board  f 

A  standard  board  is  one  which  is  12  inches  wide, 
one  inch  thick  and  12  feet  long  ;  hence,  a  standard 
board  is  one  inch  thick  and  contains  12  square  feet. 

What  is  a  standard  saw  log  f 

A  standard  log  is  12  feet  long  and  24  inches  in 
diameter. 

How  will  you  find,  the  number  of  feet  of  boards 
which  can  be  sawed  from  a  standard  log  ? 

If  we  saw  off,  say  two  inches  from  each  side,  the 
log  will  be  reduced  to  a  square  20  inches  on  a  side. 
Now,  since  a  standard  board  is  one  inch  in  thickness, 
and  since  the  saw  cuts  about  one  quarter  of  an  inch 
each  time  it  goes  through,  it  follows  that  one  fourth 
of  the  log  will  be  consumed  by  the  saw.  Hence  we 
shall  have  20X|^the  number  of  boards  cut  from  the 
log.  Now,  if  the  width  of  a  board  in  inches  be  divided 
by  12,  and  the  quotient  be  multiplied  by  the  length 
in  feet,  the  product  will  be  the  number  of  square  feet 
in  the  board.  Hence,  ffX^^^g^^  of  log  in  feet=the 
square  feet  in  each  board.  Therefore,  20XIXffX 
length  of  log=the  square  feet  in  all  the  boards=20X 
lOXIXAX  length  of  log  =:  20X10  XIX  length. 
And  the  same  may  be  shown  for  a  log  of  any 
length. 

What,  then,  is  the  rule  for  finding  the  number  of 
feet  of  boards  which  can  be  cut  from  any  log  whatever  ? 

From  the  diameter  of  the  log  in  inches  subtract  four 


COMMERCIAL    ARITHMETIC.  81 

for  the  slabs;  then  multiply  the  remainder  by  half 
itself  and  the  product  by  the  length  of  the  log  in 
feet  and  divide  the  result  by  eight;  the  quotient  will 
the  number  of  square  feet. 

Example  1. — What  is  the  number  of  feet  of  boards 
which  can  be  cut  from  a  standard  log  ? 


Diameter, 
For  slabs, 

24  inches, 
4 

Remainder, 
Half  remainder, 

20 
10 

Length  of  log. 

200  ' 
12 

8)2400 

300=the  number  of  feet. 

2.  How  many  feet  can  be  cut  from*  a  log  12  inches 
in  diameter  and  12  feet  long  ?     Ans.  48. 

3.  How  many  feet  can  be  cut  from  a  log  20  inches 
in  diameter  and  16  feet  long  ?     Ans.  25G.. 

4.  How  many  feet  can  be  cut  from  a  log  24  inches 
in  diameter  and  16  feet  long?     Ans.  400. 

5.  How  many  feet  can  be  cut  from  a  log  28  inches 
in  diameter  and  14  feet  long  ?     Ans.  504. 


8'2  THE    MERCHANT3    AND    MECHANICS'' 


CARPENTERS  AND    JOINERS'  WORK. 


In  what  does  carpenters  and  joiners^  work  consist? 

Carpenters  and  joiners'  work  is  that  of  flooring, 
roofing,  etc.,  £«id  is  generally  measured  by  the  square 
of  100  square  feet. 

When  is  a  roof  said  to  have  a  true  pitch  f 

In  carpentry  a  roof  is  said  to  have  a  true  pitch 
when  the  length  of  the  rafters  is  three  fourths  the 
breadth  of  the  building.  The  rafters  then  are  nearly 
at  right  angles.  It  is,  therefore,  customary  to  take 
once  and  a  half  times  the  area  of  the  flat  of  the  build- 
ing for  the  area  of  the  roof. 

Example  1. — How  many  squares,  of  100  square  feet 
each,  in  a  floor  48  feet  6  inches  long  and  24  feet  3 
inches  broad?     Ans.  11  and  76|  sq.  ft. 


2.  A  floor  is  36  feet  6  inches  long  and  16  feet  6 
inches  broad,  hiow  many  squares  does  it  contain  ? 
Ans.  5  and  98|  sq.  ft. 

3.  How  many  squares  are  there  in  a  partition  91 
feet  9  inches  long  and  11  feet  3  inches  high  ?  Ans. 
10  and  32  sq.  ft. 

4.  If  a  house  measure  within  the  walls  52  feet  8 
inches  in  length  and  30  feet  6  inches  in  breadth,  and 
the  roof  be  of  the  true  pitch,  what  will  the  roofing 
cost  at  11.40  per  square  ?     Ans.  $33,733. 

Of  Bins  for  Grain. 
What  is  a  bin  ?  ^ 

It  is  a  wooden  box  used  by  farmers  for  the  storage 
of  their  grain. 


COMMERCIAL    ARITHMETIC.  83 


Of  what  are  bins  generally  made  ? 

Their  bottoms  or  bases  are  generally  rectangles 
and  horizontal  and  their  sides  vertical. 

How  many  cubic  feet  are  there  zVi  a  bushel  ? 

Since  a  bushel  contains  2150.4  cubic  inches,  and 
a  cubic  foot  1728  inches,  it  follows  that  a  bushel 
contains  1\  cubic  feet,  nearly. 

Having  any  number  of  bushels,  how  then  will  you 
find  the  corresponding  number  of  cubic  feet  ? 

Increase  the  number  of  bushels  one  fourth  itself, 
and  the  result  will  be  the  number  of  cubic  feet. 

Example  1. — A  bin  contains  3T2  bushels  ;  how 
many  cubic  feet  does  it  contain  ? 

372-f-4=93;  hence,  3t2-|-93=465  cubic  feet. 

2.  In  a  bin  containing  400  bushels  how  many 
cubic  feet  ?     Ans.  500. 

How  will  you  find  the  nuniber  of  bushels  which  a  bin 
of  a  given  size  will  hold  ? 

Find  the  content  of  the  bin  in  cubic  feet,  then 
diminish  the  content  by  one  fifth,  and  the  result  will 
b«  the  content  in  bushels. 

3.  A  bin  is  8  feet  long,  4  feet  wide  and  5  feet  high ; 
how  many  bushels  will  it  hold  ? 

8X4X5=160 
then,   160-j-5=  32  :  160  —  32=128   bushels= 
capacity  of  bin. 

4.  How  many  bushels  will  a  bin  contain  which  la 


84  THE    MERCIIAXTS    AXD    MECHANICS' 

7  feet  loiio",  3  feet  wide  and  6  feet  in  height.     Ans. 
100.8  bush. 

How  ivill  ipu  find  the  dimensions  of  a  bin  which 
shall  contain  a  given  number  of  bushels  ? 

Increase  the  number  of  bushels  one  fourth  itseli 
and  the  result  will  show  the  number  of  cubic  feet 
which  the  bin  will  contain.  Then,  when  the  two 
dimensions  of  the  bin  are  known,  divide  the  last 
result  by  their  product,  and  the  quotient  will  be  the 
other  dimension. 

5.  What  must  be  the  height  of  a  bin  that  will  con- 
tain 600  bushels,  its  length  being  8  feet  and  its 
breadth  4  ? 

600-^-4=150  ;  hence,  600+150=750=:thc  cubic 
feet,  and  8X^=32,  the  product  of  the  given  dimen- 
sions. Then,  750-^32=23.44  feet  the  height  of  the 
bin. 

6.  What  must  be  the  width  of  a  bin  that  shall 
contain  900  bushels,  the  height  being  12  and  the 
length  10  feet? 

900-^4=225;  hence,  900+225=1 135=the  cubic 
feet ;  and  12X10=120,  the  product  of  the  given 
dimensions.  Then,  1125-^120=9.315  feet,  the  width 
of  the  bin. 

7.  The  length  of  a  bin  is  4  feet,  its  breadth  5  feet 
6  inches;  what  must  be  its  height  that  it  may  con- 
tain 136  bushels  ?     Ans.  7  ft.  8  in.+ 

8.  The  depth  of  a  bin  is  6  feet  2  inches,  the  breadth 
4  feet  8  inches;  what  must  be  the  length  thwi  it  may 
contain  200  bushels?     Ans.  104  in.-j- 


COM>IERCIAL    ARITHMETIC. 


85 


SLATERS  AND  TILERS'  WORK. 
How  %s  the  content  of  a  roof  found  f 

In  this  work  the  content  of  the  roof  is  found  by 
multiplying  the  length  of  the  ridge  by  the  girt  from 
.cave  to  eave.  Allowances,  however,  must  be  made 
for  the  double  rows  of  slate  at  the  bottom. 

Example  1. — The  length  of  a  slated  roof  is  45  feet 
9  inches,  and  its  girt  34  feet  3  inches;  what  is  its 
content  ?     Ans.  1566.9375  sq.  ft. 

2.  What  will  the  tiling  of  a  bam  cost  at  $3.40  per 
square  of  100  feet,  the  length  being  43  feet  10  inches 
and  breadth  27  feet  5  inches  on  the  flat,  the  eave 
board  projecting  16  inches  on  each  side  and  the  roof 
being  of  the  true  pitch  ?     Ans.  $65.26. 

BRICKLAYERS'  WORK. 

In  how  many  ways  is  artificers^  work  computed  ? 

Artificers'  work  in  general  is  computed  by  three 
dififerent  measures,  viz  : 

1st.  The  linear  measure;  or,  as  it  is  called  by  me- 
chanics, running  measure. 

2d.  Superficial  or  square  measure,  in  which  the 
cojnputation  is  made  by  the  square  foot,  square 
yard,  or  by  the  square  containing  100  square  feet  or 
yards. 

3d.  By  the  cubic  or  solid  measure,  when  it  is  esti- 
mated by  the  cubic  foot  or  the  cubic  yard.  The 
work,  however,  is  often  estimated  in  square  mea- 
sure, and  the  materials  for  construction  in  cubic 
measure. 


86  THE    MERCHANTS    AND    MECHANICS' 

What  proportion  do  the  dimensions  of  a  brick  bear 
to  each  other  f 

The  dimensions  of  a  brick  generally  bear  the  fol- 
lowing proportions  to  each  other,  viz  : 

Length=twice  the  width,  and 
Width=twice  the  thickness; 

and,  hence,  the  length  is   equal   to  four  times  the 
thickness. 

What  are  the  common  dimensions  of  a  brick  ?  How 
many  cubic  inches  does  it  contain  ? 

The  common  length  of  a  brick  is  8  inches,  in  which 
case  the  width  is  4  inches  and  the  thickness  2  inches. 
A  brick  of  this  size  contains  8X4X2^^64  cubic 
inches  ;  and  since  a  cubic  foot  contains  1728  cubic 
inches,  we  have  1728-f-64==2t  the  number  of  bricks 
in  a  cubic  foot. 

If  a  brick  is  9  inches  long,  what  will  be  its  width 
and  what  its  content  ? 

If  the  brick  is  9  inches  long,  then  the  width  is 
4 J  inches,  and  the  thickness  2|;  and  then  each  brick 
will  contain  9XHXH=^H  cubic  inches;  and  1728 
-^91 1=19  nearly,  the  number  of  bricks  in  a  cubic 
foot.  In  the  examples  which  follow  we  shall  s  n- 
pose  the  brick  to  be  8  inches  long. 

How  do  you  find  the  number  of  bricks  re  ired 
to  build  a  wall  of  given  dimensions  ? 

1st.  Find  the  content  of  the  wall  in  cubic  i^et. 

2d.  Multiply  the  number  of  cubic  feet  by  the  num- 
ber of  bricks  in  a  cubic  foot,  and  the  result  will  be 
the  number  of  bricks  required. 


COMMERCIAL    ARITHMETIC.  87 

Example  1. — How  many  bricks,  of  8  inches  in  length, 
will  be  required  to  build  a  wall  30  feet  long,  a  brick 
and  a  half  thick  and  15  feet  in  height  ?     Ans.  12150. 

2.  How  many  bricks,  of  the  usual  size,  will  be  re- 
quired to  build  a  wall  50  feet  long,  2  bricks  thick 
and  36  feet  in  height  ?     Ans.  64800. 

What  allowance  is  made  for  the  thickness  of  the  mor- 
tar? 

The  thickness  of  mortar  between  the  courses  is 
nearly  a  quarter  of  an  inch,  so  that  four  courses  will 
give  nearly  one  inch  in  height.  The  mortar,  there- 
fore, adds  nearly  one  eighth  to  the  height;  but  as 
one  eighth  is  rather  too  large  an  allowance,  we  need 
not  consider  the  mortar  which  goes  to  increase  the 
length  of  the  wall. 

3.  How  many  bricks  would  be  required  in  the  first 
and  second  examples,  if  we  make  the  proper  allow- 
ance for  mortar  ?     Ans.  1st.  106314.     2d.  56700. 

How  do  bricklayers  generally  estimaie  their  work  ? 

Bricklayers  generally  estimate  their  work  at  so 
much  per  thousand  bricks. 

4.  What  is  the  cost  of  a  wall  60  feet  long,  20  feet 
high  and  2^  bricks  thick  at  ^7.50  per  thousand — 
which  price  we  suppose  to  include  the  cost  of  the 
mortar  ? 

If  we  suppose  the  mortar  to  occupy  a  space  equal 
to  one  eighth  the  height  of  the  wall,  we  must  find 
the  quantity  of  bricks  under  the  supposition  that 
the  wall  was  17J  feet  in  height.     Ans.  $354.37^. 

In  estimaling  the  bricks  for  a  house  what  allowances 
are  made  f 


88  THE    MERCHANTS   AND    MECHANICS^ 

In  estimating  the  bricks  for  a  house,  allowance 
must  be  made  for  the  windows  and  doors. 

Of  Cisterns. 

What  are  cisterns  f 

Ciaeerns  are  large  reservoirs  constructed  to  hold 
water;  and,  to  be  permanent,  should  be  made  either 
of  brick  or  masonry.  It  frequently  occurs  that  they 
are  to  be  so  constructed  as  to  hold  given  quantities 
of  water,  and  then  it  becomes  a  useful  and  practical 
problem  to  calculate  their  exact  dimensions. 

JIoio  many  cubic  inches  in  a  hogshead  ? 

A  liogshead  contains  63  gallons,  and  a  gallon  con- 
tains 231  cubic  inches;  hence,  231X63X14553,  the 
number  of  cubic  inches  in  a  hogshead. 

How  do  youjind  the  number  of  hogsheads  which  a 
cistern  of  gixjcn  dimensions  will  contain  ? 

1st.  Find  the  solid  content  of  the  cistern  in  cubic 
inches. 

2d.  Divide  the  content  so  found  by  14553  and  the 
quotient  will  be  the  number  of  hogsheads. 

Example. — The  diameter  of  a  cistern  is  6  feet  6 
inches,  and  height  10  feet ;  how  many  hogsheads 
docs  it  contain  ? 

The  dimensions  reduced  to  inches  are  *I8  and  120; 
then,  the  content  in  cubic  inches,  which  is  5*13404.- 
832,  gives 

573404.832-f-14553=39.40  hogsheads,  nearly. 

If  the  height  of  a  cistern  be  given  how  do  you  find 
the  diameter,  so  that  the  cistern  shall  contain  a  given 
number  of  hogsheads  ? 


COMMERCIAL    ARITHincTlO,  89 

1st.  Reduce  the  height  of  the  cistern  to  inches, 
and  the  content  to  cubic  inches. 

2d.  Multiply  the  height  by  the  decimal  .t854. 

3d,  Divide  the  content  by  the  last  result  and  ex- 
tract the  square  root  of  the  quotient,  which  Avill  be 
the  diameter  of  the  cistern  in  inches. 

Example. — The  height  of  a  cistern  is  10  feet;  what 
must  be  its  diameter  that  it  may  contain  40  hogs- 
heads?    Ans.  *I8.6  in.  nearly. 

If  the  diameter  of  a  cistern  be  given  how  do  you  find 
the  height,  so  that  the  cistern  shaU  contain  a  given  num- 
ber of  hogsheads  ? 

1st.  Reduce  the  content  to  cubic  inches. 

2d.  Reduce  the  diameter  to  inches,  and  then  mul- 
tiply its  square  by  the  decimal  .1854. 

3d.  Divide  the  content  by  the  last  result,  and  the 
quotient  will  be  the  height  in  inches. 

Example. — The  diameter  of  a  cistern  is  8  feet;  what 
must  be  its  height  that  it  may  contain  150  hogs- 
heads ?    Ans.  25  ft.  1  in.,  nearly. 

MASONS'  WORK. 

What  belongs  to  masonry,  and  what  measures  are 
used? 

All  sorts  of  stone  work.  The  measure  made  use 
of  is  cither  superficial  or  solid. 

Walls,  columns,  blocks  of  stone  or  marble  are  mea- 
sured by  the  cubic  foot;  and  pavements,  slabs,  chim- 
ney pieces,  etc.,  are  measured  by  the  square  or 
superficial  foot.     Cubic  or  solid  measure  is  always 


90'  THE    MERCIIAXTJ    AND    MECHANICS' 

used   for  the  materials,  and  the  square  measure  is 
sometimes  used  for  the  workmanship. 

Example  1. — Required,  the  solid  content  of  a  wall 
53  feet  6  inches  long,  12  feet  3  inches  high  and  2 
feet  thick.     Ans.  1310J  ft. 

2.  What  is  the  solid  content  of  a  wall,  the  length 
of  which  is  24  feet  3  inches,  height  10  feet  9  inches, 
and  thickness  2  feet  ?     Ans.  521.375  ft. 

3,  In  a  chimney-piece  we  find  the  following  dimen- 
sions : 

Length  of  the  mantel  and  slab,  4  feet  2  inches. 
Breadth  of  both  together,  3    "    2 

Length  of  each  jam,  4    "    4       " 

Breadth  of  both,  1    "    9       " 

Required,  the  superficial  content.     Ans.  31  ft.  10'. 

PLASTERERS'  WORK. 

How  many  kinds  of  plasterera^  work  are  there,  and 
how  are  they  measured  ? 

Plasterers'  work  is  of  two  kinds,  viz  :  ceiling, 
which  is  plastering  on  laths;  and  rendering,  which 
is  plastering  on  walls.  These  are  measured  sepa* 
rately. 

The  contents  are  estimated  either  by  the  square 
foot,  the  square  yard,  or  the  square  of  100  feet. 

Inriched  mouldings,  etc.,  are  rated  by  the  running 
or  lineal  measure. 

In  estimating  plastering,  deductions  are  made  for 
chimneys,  doors,  windows,  etc. 

Example  1. — How  many  square  yards  are  con- 
tained in  a  ceiling  43  feet  3  inches  long  and  25  feet 
«  inches  broad  ?     Ans.  122|-,  nearly. 


COMMERCIAL   ARITHMETIC.  91 

2.  What  is  the  cost  of  ceiling  a  room  21  feet  8 
inches  bv  14  feet  10  inches,  at  18  cents  per  square 
yard  ?     Ans.  $6.42f 

3.  The  length  of  a  room  is  14  feet  5  inches, 
breadth  13  feet  2  inches,  and  height  to  the  under 
side  of  the  cornice  9  feet  3  inches.  The  cornice  girts 
8|  inches  and  projects  5  inches  from  the  wall  on  the 
upper  part  next  the  ceiling,  deducting  only  for  one 
door  7  feet  by  4;  what  will  be  the  amount  of  the 
plastering  ? 

(  53  yds.  5  ft.  3'  6"  of  rendering. 
Ans.  ^18  yds.  5  ft.  6'  4"  of  ceiling. 
(37  ft.  10' 9"  of  cornice. 

How  is  the  area  of  the  cornice  found  in  the  above 
examples  f 

The  mean  length  of  the  cornice  both  in  the  length 
and  breadth  of  the  house  is  found  by  taking  the  mid- 
dle line  of  the  cornice.  Now,  since  the  cornice  pro- 
jects 5  inches  at  the  ceiling,  it  will  project  2|  inches 
at  the  middle  line  ;  and,  therefore,  the  length  of  the 
middle  line  along  the  length  of  the  room  will  be  14 
feet,  and  across  the  room  12  feet  9  inches.  Then 
multiply  the  double  of  each  of  these  numbers  by  the 
girth,  which  is  8|  inches,  and  the  sum  of  the  pro- 
ducts will  be  the  area  of  the  cornice. 

PAINTERS'  WORK. 

How  IS  painter^  work  computed,  and  what  allow- 
ances are  made  f 

Painters'  work  is  computed  in  square  yards.  Every 
part  is  measured  where  the  color  lies,  and  the  mea- 
suring line  is  carried  into  all  the  mouldings  and  cor- 
nices. 


92  THE    MERCHANTS    AND    MECHANICS* 

Windows  are  generally  done  at  so  much  a  piece. 
It  is  usual  to  allow  double  measure  for  carved  mould- 
ings, etc. 

Example  1. — How  many  yards  of  painting  in  a 
room  which  is  65  feet  6  inches  in  perimeter  and  12 
feet  4  inches,  in  height  ?     Ans.  89^^  sq.  yds. 

2.  The  length  of  a  room  is  20  feet,  its  breadth  14 
feet  6  inches,  and  height  10  feet  4  inches;  how 
many  yards  of  painting  are  in  it — deducting  a  fire- 
place of  4  feet  by  4  feet  4  inches,  and  two  windows, 
each  6  feet  by  3  feet  2  inches.     Ans  73^\  sq.  yds. 

PAVERS'  WORK. 

How  is  pavers^  work  estimated  ? 

Pavers'  work  is  done  by  the  square  yard,  and  the 
content  is  found  by  multiplying  the  length  and 
breadth  together. 

Example  1. — ^What  is  the  cost  oft  paving  a  side- 
walk, the  length  of  which  is  35  feet  4  inches  and 
breadth  8  feet. 3  inches,  at  54  cents  per  square  yard  ? 
Ans.  $n.48  9. 

2.  What  will  be  the  cost  of  paving  a  rectangular 
court  yard,  whose  length  is  63  feet  and  breadth  45 
feet,  at  28.  Q>cl.  per  square  yard — there  being,  how- 
ever, a  walk  running  lengthwise  5  feet  3  inches 
broad  which  is  to  be  flagged  with  stone  costing  3s. 
per  square  yard  ?     Ans.  ^£40  5s.  10^^. 

PLUMBERS'  AVORK. 

Plumbers'  work  is  rated  at  so  much  a  pound,  or 
else  by  the  hundred  weight.  Sheet  lead,  used  for 
gutters,  etc.,  weighs  from  6  to  12  pounds  per  square 


COMMERCIAL    ARITmiETIC. 


93 


^ot.     Leaden  pipes  vary  in  weight  according  to  the 
4iamerer  or*  their  bore  and  thickness. 

The  following  table  shows  the  weight  of  a  square 
^oot  of  sheet  lead,  according  to  its  thickness;  and  the 
>ominon  weight  of  a  yard  of  leaden  pipe,  according 
Vo  the  diameter  of  the  bore: 


Thickness 
OF  Lead. 

PousDs  TO  A  Square 
Foot. 

Bore  op 
Leaden  Pipes. 

Pounds 
peu  Yakd. 

Inch. 

5.899 

Inch. 

Of 

10 

i 

G.554 

1 

12 

i 

7.3T3 

H 

16 

\ 

8.427 

H 

18 

i 

9.831 

ij 

21 

s 

1I.T9T 

2 

24 

Example  1. — What  weight  of  lead  of -j^ij-  of  an  inch 
in  thickness  will  cover  a  flat  15  feet  6  inches  long 
and  10  feet  3  inches  broad,  estimating  the  weight 
at  6  lbs-  per  square  foot  ?     Ans.  8  cwt.  2  qr.  1 J  lb. 

2.  What  will  be  the  cost  of  130  yards  of  leaden 
pipe  of  an  inqli  and  a  half  bore,  at  8  cents  per  pound, 
-apposing  each  yard  to  weigh  18  lbs.  ?  Ans.  $187.20. 

3.  The  lead  used  for  a  gutter  is  12  feet  5  inches 
long  and  1  foot  3  inches  broad,  what  is  its  weight, 

ippo.sing  it  to  be  ^  of  an  inch  in  thickness  ?  Ans. 
:tjl  lbs.  12  oz.  13.6  dr. 


94 


THE    MERCHANTS    AND    MECHANICS' 


4.  What  is  the  weight  of  96  yards  of  leaden  pipe 
of  an  inch  and  a  quarter  bore?  Ans.  13  cwt.  2  qr. 
24  lbs. 

5.  What  will  be  the  cost  of  a  sheet  of  lead  16  feet 
6  inches  long  and  10  feet  4  inches  broad  at  5  cents 
per  pound;  the  lead  being  -J-  of  an  inch  in  thickness  ? 
Ans.  83.81. 

PERPETUAL    CALENDAR. 

To  tell  on  what  day  of  the  week  any  date  will 
transpire  for  the  period  of  three  thousand  years  from 
the  Christian  Era. 

TABLE    OF    CENTENNIAL   RATIOS. 


200,    900,  1800,  2200, 

2600.  3000,  ratio  is 

0 

300,  1000,                                         ■     u      u 

6 

400,  1100,  1900,  2300,  2100,               "      " 

5 

500,  1200,  1600,  2000,  2400,               "      " 

4 

600,  1300,                                             ''      " 

3 

100,  1400,  1100,  2100,  2500,  2900,     "      " 

2 

100,    800,  1500,                                   "   •  " 

1 

TABLE    OF    MONTHLY    RATIOS. 

Ratio  of  January  is      3 

Ratio  of  July  is 

2 

"      "    February,       6 
"      "   March,            6 
"      "   April,              2 
"      "   May,               4 

"      "  August, 
"      "  September, 
"      "  October, 
"      "  November, 

5 

1 
3 
6 

"      "   June,              0 

"      "  December, 

I 

Remark. — In  Leap  Year  the  ratio  in  January  is  2, 
and  of  February,  5.  The  ratio  of  the  other  months 
remain  the  same. 

Explanation. — To  the  given  year  add  its  fourth 
pai't,  rejecting  the  fractions.     To  this  sum  add  th« 


COMMERCIAL    ARITHMETIC.  95 

day  of  the  month,  then  add  the  ratio  of  the  century 
and  the  ratio  of  tl>e  month.  Divide  this  sum  by  t; 
the  remainder  is  the  day  of  the  week,  counting  Sun- 
day as  the  first,  Monday  the  second,  etc.  Of  course, 
Saturday  being  the  seventh  day  the  remainder  will 
be  a  cipher. 

Example   1. — Required,  the   day  of  the    w*eek  for 
the  4th  of  July,  1870: 

To  the  given  year,  which  is TO 

Add  its  fourth  part,  rejecting  fractions,  .  .  17 
Add  the  day  of  the  month,  which  is  .  .  4 
Add  the  ratio  of  the  century,  1800,  which  is  0 
Add  the  ratio  of  the  month,  July,  which  is  .     2 

Divide  by 7)93 


13-2 
We  have  2  remainder,  or  the  2d  day  of  the  week, 
which  is  Monday. 

2.  The  Declaration  of  Independence  was  signed 
July  4th,  1776.     What  was  the  day  of  the  week  ? 

To  the  given  year,  which  is 76 

Add  its  fourth  part,  rejecting  fractions,  .  .  19 
Add  the  day  of  the  month,  which  is  .  .  .  4 
Add  the  ratio  of  the  century,  1700,  which  is  2 
Add  the  ratio  of  the  month,  July,  which  is   ,     2 

Divide  by    .     .    - 7)103 


14-5 
We  have  5  remainder,  or  Thursday,  the  5th  day  of 
the^week. 

3.  Upon  what  day  of  the  week  will  happen  the  1st 
of  January,  or  New  Years,  2000  f 


96 


THE    MERCHANTS    AXT>    MECHANICS' 


To  the  given  year,  whicli  is 00 

Add  its  fourth  part,  which  is  .  .  .  ^  .  .  0 
Add  the  day  of  the  month,  which  is  \  .  .  1 
Add  the  ratio  of  2000,  which  is  ...  .  4 
Add  the  ratio  of  January,  being  leap  year,  is  2 
Divide  by 7)7 

The  cipher  remaining  gives  Saturday  as  the  answer. 
4.  On  what  day  were  you  born  ? 


To 


January. . . 
February.. 

March 

April 

May 

June 

July 

August 

September 
October... 
November. 
December. 


151 
120 
92 
61 1 
311 
SGSJ 


181 
150 
122 
91 
61 
30 
335i  365 
304  334 
2731  303 
243  273 
212'  242 
182!  212 


335 


334 
303 
275 
244 
214 
163 
153 
122 
91 
61 
30 
865 


Table    showing    Difference  of  Time  at    12 
O'clock  (Noon)  at  Ne^Ar  York. 


New  York....  12.00  m. 

Buffalo 11.40  a.m. 

Cincinnati ...  .11.18  " 

Chicago 11.07  " 

St.  Louis 10.55  " 

San  Francisco.  8.45  " 

New  Orleans..  10.56  " 

Washington... 11. 48  " 

Charleston 11.36  " 

Havana 11.25  " 


Boston 12.12  p.  m. 

Quebec 12.12     " 

Portland....  12.15     " 

London 4.55     " 

Paris 5.05     " 

Rome 5.45     " 

Constantin'ple  6.41     •" 

Vienna 6.00     " 

St.Petersb'g.    6.57     " 
Pekin 12.40  a.  m. 


COMMERCIAL   ARITHMETIC.  $7 


WEIGHTS  AND  MEASURES. 

Troy  Weight. 
By  this  weight  gold,  silver,  platina  and  precious 
Ftones  (except  diamonds)  are  estimated. 

20  mites.  ...1  grain.  I  20  pennyw'ts 1  ounce. 

20  grains ....  1  pennyw't.   |  12  ounces 1  pound. 

Any  quantity  of  gold  is  supposed  to  be  divided 
into  24  parts,  called  carats.  If  pure,  it  is  said  to  be 
24  carats  fine;  if  there  is  22  parts  of  pure  gold  and  2 
parts  of  alloy  it  is  said  to  be  22  carats  fine.  The 
standard  of  American  coin  is  nine  tenths  pure  gold, 
and  is  worth  $20.67.  What  is  called  the  neiv  stan- 
dard, used  for  watch  cases,  etc.,  is   18  carats  fine. 

The  term  carat  is  also  applied  to  a  weight  of  3i 
grains  troy,  used  in  weighing  diamonds;  it  is  divided 
into  4  parts  called  grains;  4  grains  troy  are  thus 
equal  to  5  grains  diamond  weight. 

Apothecaries'  Weight,  used  in  Medical 

Prescriptions. 
The  pound  and  ounce  of  this  weight  are  the  same 
as  the  pound  and  ounce  troy,  but  differently  divided. 


8  Drachms..  1  ounce  troy 
12  ounces..... 1  pound  " 


20  grains  troy..l  scruple. 
3  scruples 1  drachm. 

Druggists  buy  their  goods  by 

Avoirdupois  Weight. 

By  this  weight  all  goods  are  sold  except  those 
named  under  troy  weight. 

2*1^1  grains 1  drachm 

16  drachms 1  ounce. 

16  ounces 1  pound. 


98  THE  MERCHANTS  AND  MECHANICS' 

28  pounds. , 1  quarter. 

4  quarters 1  hundred  weight. 

20  hundred  weight 1  ton. 

The  grain  avoirdupois,  though  never  used,  is  the 
same  as  the  grain  in  troy  weight,  ^,000  grains 
make  the  avoirdupois  pound,  and  5,T60  grains  the 
troy  pound.  Therefore,  the  troy  pound  is  less  than 
the  avoirdupois  pound  in  the  proportion  of  14  to  IT, 
nearly;  but  the  troy  ounce  is  greater  than  the  avoir- 
dupois ounce  in  the  proportion  of  T9  to  T2,  nearly. 
In  times  past  it  was  the  custom  to  allow  112  pounds 
for  a  hundred  weight,  but  usage,  as  well  as  the  laws 
of  a  majority  of  the  States,  at  the  present  time  call 
100  pounds  a  hundred  weight. 

Apothecaries*  Fluid  Measure. 

60  minims 1  fluid  drachm. 

8  fluid  drachms 1  ounce  (troy). 

16  ounces  (troy) 1  pint. 

8  pints 1  gallon. 

Measure  of  Capacity  for  all  Liquors. 

5  ounces,  avoirdupois,  of  water  make  1  gill. 

4  gills. ...  1  pint    =  34|  cubic  inches,  nearly. 

2  pints. . .  1  quart  =  69^  do. 

4  quarts . .  1  gallon  =27 1 J  do. 

31 J  gallons 1  barrel. 

42  gallons 1  tierce. 

63  gallons,  or  2  bbls 1  hogshead. 

'       2  hogsheads 1  pipe  or  butt. 

2  pipes 1  ton. 

The  gallon  must  contain  exactly  10  pounds  avoir- 
dupois of  pure  water  at  a  temperature  of  62°,  the 
barometer  being  at  30  inches.  It  is  the  standard 
unit  of  measure  of  capacity  for  liquids  and  dry  goods 


CO>nfERCTAT.    ARITHMETIO.  ^9 

of  CTory  description,  and  is  ^  larger  than  the  old 
wine  measure,  ^j  larger  than  the  old  dry  measure, 
and  "bV  ^^^^'^  t^^^^  t^^^  0^^  ^1®  measure.  The  A\ine 
gallon  must  contain  231  cubic  inches. 

Measure  of  Capacity  for  all  Dry  Goods. 

4  gills 1  pint       =     34 1  cub.  in.,  nearly. 

2  pints 1  quart    =     G9|  cubic  inches. 

4  quarts 1  gallon   =  211\  cubic  inches. 

2  gallons 1  peck      =:  554|  cubic  inches. 

4  pecks,  or  8  gals.  1  bushel  =2150|  cubic  inches. 
8  bushels 1  quarter  =     10|  cubic  ft.,  nearly. 

When  selling  the  following  articles  a  barrel  weighs 
as  hjere  stated  : 

For  rice,  600  lbs.;  flour,  196  lbs.;  powder,  25  lbs.; 
corn,  as  bought  and  sold  in  Kentucky,  Tennessee, 
etc.,  5  bushels  of  shelled  corn;  as  bought  and  sold 
at  New  Orleans,  a  flour  barrel  full  of  ears;  potatoes, 
as  sold  in  New  York,  a  barrel  contains  2 J  bushels; 
poi"k,  a  barrel  is  200  lbs.,  distinguished  in  quality  by 
"clear,"  "mess,"  "prime;"  a  barrel  of  beef  is  the 
same  weight. 

TiSe  legal  bushel  of  America  is  the  old  Winchester 
mea&Hire  of  2,150.42  cubic  inches.  The  imperial 
bushel  of  England  is  2,218.142  cubic  inches,  so  that 
32  Erglish  bushels  are  about  equal  to  33  of  ours. 

Although  we  are  all  the  time  talking  about  the 
price  of  grain,  etc.,  by  the  bushel,  we  sell  by  weight, 
as  follows: 

Wheat,  beans,  potatoes  and  clover  seed,  60  lbs.  to 
the  bushel;  corn,  rye,  flax  seed  and  onions,  66  lbs.; 
corn  on  the  cob,  70  lbs.;  buckwheat,  52  lbs.;  barley, 
48  lbs.;  hemp  seed,  44  lbs.;  timothy  seed,  45  lbs.; 
castor  beans,  46  lbs.;  oats,  35  lbs.;  bran,  20  lbs.; 
blue  grafig  se^d,   14  lbs.;  salt,  the  real  weight  of 


100  THE    MERCHANTS    AXn    MECHANICS* 

coarse  salt  is  85  lbs.;  dried  apples,  24  lbs.;  dried 
peaches,  33  lbs.,  accordlnc^  to  some  rules,  but  others 
arc  22  lbs.  per  bushel,  while  in  Indiana  dried  npplcs 
and  peaches  are  sold  by  the  heaping  bushel;  r-o  arc 
potatoes,  turnips,  onions,  apples,  etc.,  and  in  KonH"; 
sections  oats  are  heaped.  A  bushel  of  corn  in  tlie 
car  is  three  heaped  half  bushels,  or  four  even  fiiil. 

In  Tennessee  a'  hundred  ears  of  corn  is  sometimes 
counted  as  a  bushel. 

A  hoop  18i  inches  diameter  8  inches  deep  holds 
a  Winchester  bushel.  A  box  12  inches  .square  1 
and  1^^  deep  will  hoM  half  a  bushel.  A  heaping 
bushel  is  2,815  cubic  inches. 

Cloth  Measure. 

2|  inches 1  nail. 

4    nails 1  quarter  of  a  yard. 

4    quarters 1  yard. 

Foreign  Cloth  Measure. 

2|  quarters 1  ell  Hamburg. 

3    quarters 1  ell  Fiendish. 

6    quarters 1  ell  English. 

6    quarters 1  ell  French. 

Measure  of  Length. 

12  inches 1  foot. 

3  feet 1  yard. 

5|  yards 1  rod,  pole  or  perch. 

40  poles 1  furlong. 

8  furlongs,  or  1*160  yds.  1  mile, 
^q  J       .,  )1  degree  of  a  great  cir- 

^^  )  cle  of  the  earth. 

By  scientific  persons  and  revenue  officers  the  inch 
is  divided  into  tenths,  hundredths,  etc.     Among  me- 


COMMERCIAL   ARITHMETIC.  101 

chanics  the  inch  is  divided  into  eighths.  The  divi- 
sion of  the  inch  into  12  parts,  called  lines,  is  not  now 
in  use. 

A  standard  English  mile,  which  is  the  measure 
that  we  use,  is  5,280  feet  in  length,  1,760  yards,  or 
320  rods.  A  strip  one  rod  wide  and  one  mile  long 
U  two  acres.  By  this  it  is  easy  to  calculate  the 
quantity  of  land  taken  up  by  roads,  and  also  how 
much  is  wasted  by  fences. 

Gunter's  Chain. 

USED    FOR    LAND    MEASURE. 

7  j^„2jj-  inches 1  link. 

100  links,  or  66  feet,  or  4  poles 1  chain. 

10  chains  long  by  1   broad,   or  10  sq. 

chains 1  acre. 

80  chains  •  * 1  mile. 

Surface  Measure. 

144    square  inches 1  square  foot. 

9    square  feet 1  square  yard. 

30 J  square  yards 1  square  rod  or  perch. 

40    square  perches 1  rood. 

4    roods 1  acre. 

640    acres 1  square  mile. 

Mca.s'.irc  209  foet  on  each  side  and  you  have  a 
square  acre  within  an  inch. 

Tiio  following  gives  the  comparative  size  in  square 
yards  of  acres  in  different  countries: 

English  acre,  4,840  square  vard.-;  Scotcli,  0,150; 
Irish,  7,840;  Hamburg,  11,545;  Amsterdam, -9,722; 
Dantzic,  6,650;  France  (hectare),  11,9G0;  P  ussia 
(morgcii),  3,053. 

This  di(T"''"v^'''  ^liould  be  borne  in   mind  in  rciid- 


102 


THE    MERCHANTS    AND    MECHANICS' 


ing  of  tne  products  per  acre  in  different  countries. 
Our  land  measure  is  that  of  England. 

Government  Land  Measure. 

A  township — 36  sections,  each  a  mile  square. 

A  soctioM — 640  acres. 

A  quarter  section,  half  a  mile  square — 160  acroR. 

An  eighth  section,  half  a  mile  long,  north  raul 
south,  and  a  quarter  of  a  mile  wide — 80  acres. 

A  sixteenth  section,  a  quarter  of  a  mile  square — • 
40  acres. 

The  sections  are  all  numbered  1  to  36,  commen- 
cing at  the  northeast  corner  thus: 


6 

5 

4 

3 

2 

V  v.-  V  K 

1 

8 

9 

10 

11 

12 

18 

11 

10* 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

The  sections  are  divided  into  quarters,  which  are 
named  by  the  cardinal  points,  as  in  section  1.  The 
quarters  are  divided  in  the  same  way.  The  descrip- 
tion of  a  forty  acre  lot  would  read:  The  south  lialf  of 
the  west  half  of  the  southwest  quarter  of  section  1  in 
township  24,  north  of  range  1  west,  or  as  the  case 
might  be;  and  sometimes  will  fall  short  and  some- 
times overrun  the  number  of  acres  it  is  supposed  to 
contain. 

♦School  Bection. 


COMMERCIAL    ARITHMETIC.  108 

Square  Measure, 

FOR    CARPENTERS,    MASONS,    ETC. 

144  square  inches 1  square  foot. 

9  sq.  ft.,  or  1,296  sq.  in.l  square  yard. 

100  square  feet 1  sq.of  flooring,  roofing,&c. 

30^  square  yards 1  square  rod. 

36  square  yards 1  rood  of  building. 

Geographical  or  Nautical  Measure. 

C  feet 1  fathom, 

110  fathoms,  or  660  feet. .  1  furlong. 

6075^  feet 1  nautical  mile, 

3  nautical  miles 1  league. 

20  leag's,  or  60  geo.  miles.  1  degree. 

360  degrees The  earth's  circumference 

=24,855|  miles,  nearly. 

The  nautical  mile  is  195|^  feet  longer  than  the  com- 
mon mile. 

Measure  of  Solidity. 

1728  cubic  inches 1  cubic  foot. 

27  cubic  feet 1  cubic  yard. 

16  cubic  feet 1  cord  ft.,  or  ft.  of  wood. 

8  cord  ft.,  or  128  cub.  ft.l  cord, 
40  ft.  of  round,  or  50  ft. 


of  hewn  timber,      ^  ^  *^"' 
42  cubic  feet 1  ton  of  shipping. 

%.ngular  Measure,  or  Divisions  of  the  Circle. 

60  seconds 1  minute. 

50  minutes 1  degree. 

50  degrees 1  sign. 

30  degrees 1  quadrant. 

BSO  degrees 1  circumference. 


104  THE    MERCHANTS    AND    MECHANICS* 

Measure  of  Time. 

60  seconds   1  minute. 

GO  minutes 1  hour. 

24  liours 1  da}^ 

1  days 1  week. . 

28  davs 1  lunar  month. 

28,  29,  30  or  31  days 1  calen'r  month. 

12  calendar  months 1  year. 

3(')5  days 1  com.  year. 

866  days 1  leap  year. 

365^  days • 1  Julian  year. 

365  d.  5  h.  48  m.  49  s 1  Solar  year. 

365  d.  6  h.  9  m.  12  s 1  Siderial  year. 

Ropes  and  Cables. 

6  feet 1  fathom. 

1 20  feet 1  cable  length. 


MISCELLANEOUS   IMPOPTANT  FACTS 
ABOUT  WEIGHTS  AND  MEASURES. 

Board  Measure. 

Boards  are  sold  by  superficial  measure  at  so  much 
per  foot  of  one  inch  or  less  in  thickness,  adding  one 
fourth  to  the  price  for  each  quarter  inch  thickness 
over  an  inch. 

Grain  Measure  in  Bulk. 

Multiply  the  width  and  length  of  the  pile  totr^thec^ 
and  that  product  by  the  height,  and  divide  by  ^.15f|^ 
and  you  have  the  contents  in  bushels. 

If  you  wish  the  contents  of  a  pile  of  ears  of  ^-^vn, 
or  roots,  in  heaped  bushels,  ascertain  the  p^b'^Q 
inches  and  divide  by  2,818. 


COMMERCIAL    ARITHMETIC.  106 

A  Ton  Weight. 

In  this  country  a  ton  is  2,000  pounds.  In  most 
places  a  ton  of  hay,  etc.,  is  2,240  pounds,  and  in 
some  places  that  foolish  fashion  still  prevails  of 
weighing  all  bulky  articles  sold  by  the  ton  by  the 
"  long  weight,"  or  tare  of  12  lbs.  per  cwt. 

A  ton  of  round  timber  is  40  feet;  of  square  timber, 
54  cubic  feet. 

ANIMAL  STRENGTH. 
Men. 
The  mean  effect  of  the  power  of  a  man,  unaided  by 
a  machine,  working  to  the  best  practicable  advan- 
tage, is  the  raising  of  70  lbs.  1  foot  high  in  a  second, 
for  10  hours  in  a  day. 

Two  men,  working  at  a  windlass  at  right  angles 
to  each  other,  can  raise  70  lbs.  more  easily  than  cue 
man  can  30  lbs. 

The  result  of  observation  upon  animal  power  fur- 
nishes the  following  as  the  maximum  daily  effect: 

1.  When  the  effect  produced  varied  from  J  to  .2  of 
that  which  could  be  produced  without  v«>locity  du- 
ring a  brief  interval. 

2.  When  the  velocity  varied  from  }  to  ^  for  a  man, 
and  from  .08  to  .066  for  a  horse,  of  the  velocity  which 
they  were  capable  for  a  brief  interval,  and  not  pro- 
ducing any  effort. 

3.  When  the  duration  of  the  daily  work  varied 
from  ^  to  J  for  a  brief  interval,  during  vvhich  tiie 
work  could  be  constantly  sustained  without  preju- 
dice to  the  health  of  the  man  or  the  animals — the? 
time  not  extending  beyond  18  hours  per  day,  how- 
ever limited  may  be  the  daily  task,  so  lonj:  cs  it  rup- 
resents  a  constant  attendance  in  the  shop. 


106  THE    MERCHANTS   AND    MECHANICS^ 

By  Mr.  Field's  experiments  in  1838  the  maximum 
power  of  a  strong  man,  exerted  for  2|  minutes= 
18,000  lbs.  raised  one  foot  in  a  minute. 

A  man  of  ordinary  strength  exerts  a  force  of  30 
lbs.  for  10  hours  in  a  day,  with  a  velocity  of  2i  fcot 
in  a  second=4,500  raised  one  foot  in  a  minute=.2  of 
the  work  of  a  horse. 

A  man  can  travel,  without  a  load,  on  level  ground, 
during  8|  hours  a  day  at  the  rate  of  3.7  miles  an 
hour,  or  31^  miles  a  day.  He  can  carry  111  lbs.  11 
miles  in  a  day.  Daily  allowance  of  water  for  a  man, 
one  gallon  for  all  purposes,  and  he  requires  from  220 
to  240  cubic  feet  of  air  per  hour. 

A  porter,  going  short  distances  and  returning  un- 
loaded, can  carry  135  lbs.  t  miles  a  day.  He  can 
transport  in  a  wheelbarrow  150  lbs.  10  miles  in  a 
day. 

The  muscles  of  the  human  jaw  exert  a  force  of  534 
lbs. 

Mr.  Buchanan  ascertained  that,  in  working  a  pump, 
turning  a  winch,  in  ringing  a  hell  and  in  rowing  a 
boat  the  effective  power  of  a  man  is  as  the  numbers 
100,  16t,  227  and  248. 

A  man  drawing  a  boat  in  a  canal  can  transport 
110,000  lbs.  for  a  distance  of  7  miles,  and  produce 
156  times  the  effect  of  a  man  weighing  154  lbs.  and 
walking  31 J  miles  in  a  day.  He  can  also  produce 
an  effect  upon  a  tread-wheel  of  30  lbs.,  with  a  velocity 
of  2,3  feet  in  a  second,  for  7  hours  in  a  day,  and  can 
draw  or  push  on  a  horizontal  plane  30  lbs.  with  a 
velocity  of  2  feet' in  a  second  for  8  hours  in  a  day. 
tie  can  raise  by  a  single  pulley  38  lbs.  with  a  velo- 
nty  of  .8  of  a  foot  per  second  for  8  hours  in  a  day, 
And  he  can  pass  over  12|  times  the  space  horizon- 
tally *'hat  he  can  vertically. 


COMMERCIAL    ARITHMETIC. 


107 


A.  foot  soldier  travels  in  1  minute,  in  common  time, 
90  steps=10  yards. 

A  foot  soldier  travels  in  1  minute,  in  quick  time, 
no  f>;tops=86  yards. 

A  foot  soldier  travels  in  1  minute,  in  double-quick 
ime,  140  stepson 0  3'ards. 

lie  occupies  in  the  ranks  a  front  of  20  inches  and 
^nd  a  depth  of  13,  without  a  knapsack;  the  interval 
between  the  ranks  is  13  inches. 

Average  weight  of  men,  150  lbs.  each. 

Five  men  can  stand  in  a  space  of  1  square  yard. 


TABLE  OF  THE 


EFFECTIVE  POWER  OF  MEN  FOR  A  SHORT 
PERIOD. 


MANNEB  OF  APPUCATIOS. 

FOBCE. 

MANNER  OF  APPLICATION. 

FOBCE. 

Bcnr-hviceOTehlsel 

Brace  bit 

Drawing  knife  Or  auger 

HaTid  i>l'\D'» 

Lbs. 

72 

16 
100 

60 

36 

Screwdriver,  one  hand 

Small  screwdriver 

Thumb  and  fingers 

Lbs. 
84 
14 
14 
45 

liuud-saw 

Wiudlasa  or  piucers 

60 

Horses. 

A  hoi-ee  cnn  travel  400  yards  at  a  walk  in  4i  min- 
utes, at  a  trot  in  2  minutes,  and  at  a  gallop  in  1 
minute.  He  occupies  in  the  ranks  a  front  of  40  inches 
and  a  depth  of  10  feet;  in  a  stall,  from  3i  to  41  feet 
front;  and  at  a  picket,  3  feet  bv  9;  and  his  average 
weight=l,000  lbs. 

A  horse,  carrying  a  soldier  and  his  equipments  (225 
lbs.),  can  travel  25  miles  in  a  day  (8  hours). 

A  draught  horse  can  draw  1,600  lbs.  23  miles  a 
day,  weiglit  of  carriage  included. 

The  ordinary  work  of  a  horse  may  be  stated  at 


108 


THE   MERCHANTS   AND   MECHANICS' 


22,500  lbs.,  raised  1  foot  in  a  minute,  for  8  hours  a 
day. 

In  a  horse-mill  a  horse  moves  at  the  rate  of  3  feet 
in  a  second.  The  diameter  of  the  track  should  not 
be  less  than  25  feet. 

A  horse-power  in  machinery  is  estimated  at  33,0G0 
lbs.,  raised  1  foot,  in  a  minute;  but  as  a  horse  can 
exert  that  force  but  6  hours  in  a  day,  one  machinery 
horse-power  is  equivalent  to  that  of  4i  horses. 

The  expense  of  conveying  goods  at  3  miles  per 
hour  per  horse  teams  being  1,  the  expense  at  4| 
miles  will  be  1.33,  and  so  on — the  expense  being 
doubled  when  the  speed  is  5|  miles  per  hour 

The  strength  of  a  horse  is  equivalent  to  that  of  5 
inen. 

The  daily  allowance  of  water  for  a  horse  should 
be  4  gallons. 

TABLE  OF  THE  AMOUNT  OF  LABOR  A  HORSE  OF  AVERAGE 
STRENGTH  IS  CAPABLE  OF  PERFORMING  AT  DIFFERENT 
VELOCITIES,    ON    CANALS    RAILROADS    AND    TURNPIKES. 

Force  of  Traction  estimated  a^  83.3  lbs. 


USEFUL   EFFECT  FOB  ONE 

BAY,    DEAWN 

Velocity  peb 

DUEATION  OF 
WOEK. 

ONE   MILE. 

HOUK. 

ON  A  CANAL. 

ON  A  RAILE'r 

ON  A  T'NPIKE. 

MILES. 

HOXTBS. 

TONS. 

TONS. 

TONS. 

2}4 

11.5 

520 

115 

U 

3 

8. 

243 

92 

12 

4 

4.5 

102 

72 

9 

5 

2.9 

52 

57 

7.2 

6 

2. 

30 

48 

6 

7 

1.5 

19 

41 

6.1 

8 

1.125 

12.8 

36 

4.5 

10 

.75        • 

6.6 

28.8 

3.6 

The  actual  labor  performed  by  horses  is  greater, 
but  they  are  injured  by  it. 


COMMERCIAT.    ARITHMETIC.  109 

A  horse  in  a  mill  can  produce  an  effect  of  106  lbs. 
at  a  velocity  of  3  feet  in  a  second  for  8  hours  in  a 
day.  A  mule  can  produce,  under  a  like  velocity  and 
time,  au  efifect  of  tl  lbs.,  and  an  ass  37  lbs. 

An  ox,  walking  at  a  velocity  of  2  feet  in  a  second 
(1.34  miles  per  hour),  will  draw  154:  lbs.  for  8  hours 
i I  a  day. 

A  horse  requires  a  space  of  *I  feet  by  2^  for  trans- 
portation in  a  vessel;  and  a  beef  requires  CJ  feet  by 
26  inches,  without  manger,  and  2  feet  additional 
length  with  one.  3  beeVes  or  15  sheep  lequire  th« 
food  of  2  horses. 


no 


THE    MERCHANTS    AND    MECHANICS' 


TABLE    SHOWING    THE    AMOUNT    OP     LABOR     PRODUCED     BY 
ANIMAL   POWER   UNDER   DIFFERENT    CIRCUMSTANCES. 


MANNER  OP  APPLICATION. 


Power. 

Veloci- 
ty l>er 
Second. 

Lbs. 

Feet. 

6 

ly. 

132 

5/ 

6 

2)i 

13 

2>^ 

132 
143 

1 
5 

154 
264 

3 

3X 

1540 
li>iO 

2 

3X 

143 

K 

26 
18 
140 
26 
100 
133 
66 
33 

2 

2'^ 
'A 
5 
3 
2 
3 
2X 

88 

V< 

140 

Vi 

140 

41 

.2 

66 

6% 

97 

7X 

770 

7)i 

W*'''-'*'*   ,  Power 
Minute,   j  L *;.    . 


10  Hours  pkk  Da  v. 
Jilan,  throwing  earth  ■with  a  shovel,  a  height 

of  5  feet 

Man,  vvheeling  a  loaded  barrow  up  an  in- 
clined plane,  height  one  twelfth  of  length 
)ian,  raising  and  pitching  earth  in  a  shovel 

to  a  horizontal  distance  of  13  feet — 

Jklan,  pushing  and  drawing  alternately  in  a 

vertical  direction 

JIan,  transporting  weight  upon  a  barrow, 

and  returning  unloaded 

Man,  walking  upon  a  level 

^orse,  drawing  a   four-wheeled  carriage 

at  a  walk 

Horse,  with  load  upon  his  back,  at  a  walk. 
Horse,  transporting  a  loaded  wagon,  and 

returning  unloaded  at  a  walk 

Horse,  drawing  a  loaded  wagon  at  a  walk . . 

8   n0UU3   PER  D.VY. 

Man,  ascending  a  slight  elevation,  unloaded 
Man,  walking,  and  pushing  or  drawing  in  a 

horizontal  direction 

Man,  turning  a  crank 

Man,  upon  a  tread-mill 

Man,  rowing 

Horse,  upon  a  revolving  platform,  at  a  walk 
Ox,  upon  a  revolving  platform,  at  a  walk. . . 
Mule,  upon  a  revolving  platform,  at  a  walk 
Ass,  upon  a  revolving  platform,  at  a  walk. . 

"!  Hours  per  Day. 

Man,  walking  with  a  load  upon  his  back. . . 

6  Hours  per  Day. 

Man,  transporting  a  weight  upon  his  back, 
and  returning  unloaded 

Man,  transporting  a  weight  upon  his  back 
up  a  slight  elevation,  and  returning  un- 
loaded  

Man,  raising  a  weight  by  the  hands 

4^  Hours  per  Day. 

Tlorsc,  upon  a  revolving  platform  at  a  trot. 

Horse,  drawing  an  unloaded  four-wheeled 
carriage  at  a  trot 

Hor;>e,  drawing  a  loaded  four-wheeled  car- 
riage at  a  trot 


Lbs. 

480 

4950 

810 

1950 

7020 
42'JOO 

27720 
59400 

184800 
346500 

4290 

3120 
2790 
•4200 
7800 
18000 
15840 
11880 
5280 

13200 


14700 


1320 


43195 
334950 


8.7 
90. 
14.7 
35.5 

144 

7yo 

504 
lOSO 

3360 
6300 


45.2 

39 

00.9 
113 
260.8 
229.5 
173.2 

76.5 

1G7.9 


19 

14.4 


218.7 
353.5 
2741 


COMMERCIAL   ARITHMETIC. 


Ill 


How  many  men  are  required  upon  a  tread-mill,  20 
feet  in  diameter,  in  order  to  raise  a  weight  of  900 
fbs.,  the  crank  being  9  inches  in  length  ? 

The  weight  of  the  wheel  and  its  load  is  estimated  r.t 
5000  lbs.,'and  the  friction  at  .0I5=at  ^5  lbs.  The 
labor  of  a  man  upon  such  a  mill  is  estimated  ;.t 
25  lbs.    Lenrrth  of  crank=.75  feet. 

o 

Then  900X.'«5+5000X-015=T50  lbs.,  the  resistance 
750 

of  the  icheel ;  and =75  lbs.,  the  power  required 

20-^2 
at  the  circumference  of  the  icheel. 

Therefore,  *I5-i-25=3  7nen. 

The  draught  of  man  and  animals  by  traces  is  as 
follows  : 

Man. . .  .150  lbs.    Horse 600  lbs.    Mule 500  lbs.    Ass 360  lbs. 

A  man  rowing  a  boat  1  mile  in  7  minutes  performs 
the  labor,  while  rowing,  of  6  fully  worked  laborers 
at  ordinary  occupations  of  10  hours. 

TABLE    SHOWING   THE    EFFECTS  OF  A  TRACTION  OF    100  LBS. 
AT   DIFFERENT   VELOCITIES    ON    CANALS. 


V.loelty 

TrioHly  va 

MaM 

Virfal 

Velocity 

Velocity 

Mam 

Useful 

p«Uo»r. 

B€«ondr 

MOTtd. 

Effea. 

p«  Hoar. 

per  Second. 

Moved. 

Effect. 

MIlM. 

Fe^ 

I.b.. 

Lb«. 

Wile«. 

Feet. 

T.hs. 

\M. 

2,'i 

8.66 

55500 

39406 

6 

8.8 

9635 

6840 

3 

4.4 

sa-via 

27361 

7 

10.26 

7080 

5026 

3'i 

5.13 

28:316 

20100 

8 

11.73 

5420 

3848 

4 

6.86 

21680 

15390 

9 

13.2 

4282 

3040 

5 

7.33 

18876 

9850 

10 

14.66 

S4G8 

2462 

The  load  carried,  added  to  the  weight  of  the  ves- 
pel  which  contains  it,  forms  the  total  mass  moved, 
and  the  useful  effect  is  the  load. 

The  force  of  traction  on  a  railroad  or  turnpike  is 
constant,  but  the   mechanical  power  necessary  to 


112  THE    MERCHANTS    AND    MECHANICS' 

move  the  carriage  increases  with  the  velocity.  On  t 
canal  the  force  of  traction  varies  as  the  square  of 
the  vck)city. 

"Labor  upon  Embankments. 

ELLWOOD    MOERIS. 

Single  Horse  and  Cart. — A  horse  with  a  loaded 
dirt  cart,  employed  in  excavation  and  embankment, 
will  make  100  lineal  feet  of  trip,  or  200  feet  in  dis- 
tance, per  minute,  while  moving.  The  time  lost  in 
loading,  dumping,  awaiting,  etc. ,=4  minutes  per 
load. 

A  medium  laborer  will  load  with  a  cart  in  10  hours, 
of  the  following  earths,  measured  in  the  bank: 

Gravelly  earth,  10;  loam,  12;  and  sandy  earth,  14 
cubic  yards. 

Earth  from  a  natural  excavation  occupies  i  more 
space  than  when  transported  to  an  embankment. 

Carts  are  loaded  as  follows:  Descending  hauling,  \ 
of  a  cubic  yard  in  bank;  level  haiding,  ^  of  a  cubic 
yard  in  bank;  ascending  hauling,  J  of  a  cubic  yard 
in  bank. 

Loosening,  etc. — In  loam,  a  three-horse  plow  will 
loosen  from  250  to  800  cubic  yards  per  day  of  10 
hours. 

The  cost  of  loosening  earth  to  be  loaded  will  be 
from  1  to  8  cents  per  cubic  yard  when  wages  are 
105  cents  per  day. 

The  cost  of  trimming  and  bossing  is  about  2  cents 
per  cubic  yard. 

Scooping. — A  scoop  load  will  measure  -^  of  a  cubic 
yard   measured  in  excavation. 

The  time  lost  in  loading,  unloading  and  returning, 
per  load,  is  1|  minutes. 


COMMERCIAL    ARITHMETIC. 


113 


The  time  lost  for  every  10  feet  of  distance  from 
excavation  to  bank,  and  returning,  is  1  minute. 

In  double  scooping  the  time  lost  iu  londinjr,  return- 
inpr,  etc.,  will  be  1  minute,  and  in.single  scooping  it 
will  be  IJ  minutes. 

VOLUMES  OF  EXCAVATION  AND  EMBANKMENT. 

The  volume  of  earth  in  embankment  is  less  than  in 
excavation,  as  the  compression  of  earth  in  an  embank- 
ment is  in  excess  of  the  expansion  of  its  volume  in  a 
natural  state,  the  proportion  being  as  follows:  Sand, 
i;  clay,  ^;  gravel,  J. 

The  volume  of  rock  in  bank  exceeds  that  in  exca- 
vation in  the  proportion  of  3  to  2. 

Measurement  and  Computation  of  the  Ton- 
nage of  Vessels  under  the  Act  of  Con- 
gress of  6th  of  May,  1864. 

Measurements  are  expressed  in  feet  and  decimals 
of  a  foot,  and  tonnage  in  tons  and  hundredths  of  a 
ton. 

The  "  tonnage  length"  is  the  lengtii  along  the  mid- 
dle line  of  the  vessel  upon  the  under  side  of  the  ton- 
nage deck  plank,  but,  for  convenience,  is  measured 
upon  the  top  of  the  deck,  and  is  the  length  between 
these  extremities,  which  is  divided  into  a  number  of 
parts,  according  to  the  classification  rtiade  under 
the  law. 

The  depths  are  perpendicular  and  the  breadths 
horizontal;  the  upper  breadth,  which,  in  every  case, 
passes  through  the  top  of  the  tonnage  depth,  being 
at  a  distance  below  the  deck  at  its  middle  line  equal 
to  one  third  of  the  spring  of  the  beam  at  that  point, 
and  thus  passing  through  the  deck  upon  each  .side; 
and  the  lower  breadth,  which  is  at  the  bottom  of  the 


114  THE    MERCPIANTS    AND    MECHANICS' 

tonnage  depth,  being  at  a  distance  above  the  upper 
side  of  the  floor  timber  at  the  inside  of  the  limber 
strake,  equal  to  the  average  thickness  of  the  ceiling, 
and  thus  passing  through  the  keelson. 

The  "  spring  of  the  beam''  is  the  perpendicular 
distance  from  the  crown  of  the  tonnage  deck  at  the 
centre  to  a  line  stretched  from  end  to  end  of  the 
beam,  and  must  be  ascertained  at  each  point  where 
it  is  to  be  used  in  the  measurement. 

Tlie  register  of  every  vessel  expresses  her  length 
and  breadth,  together  with  her  depth,  and  the  height 
under  the  third  or  spar  deck,  and  is  ascertained  in 
the  following  manner  :  The  tonnage  deck,  in  ves- 
sels having  three  or  more  decks  to  the  hull,  isthe 
second  deck  from  below;  in  all  other  cases  the  upper 
deck  of  the  hull  is  the  tonnage  deck.  The  length 
from  the  fore  part  of  the  outer  planking  upon  the 
side  of  the  stem  to  the  after  part  of  the  main  stern 
post  of  screw  steamers,  and  to  the  after  part  of  the 
rudder  post  of  all  other  vessels,  measured  upon  the 
top  of  the  tonnage  deck,  is  accounted  the  vessel's 
length.  The  breadth  of  the  broadest  part  upon  the 
outside  of  the  vessel  is  accounted  the  vessel's  breadth 
of  beam.  A  measure  from  the  under  side  of  the  ton- 
nage deck  plank  amidships  to  the  ceiling  of  the  hold 
(average  thickness)  is  accounted  the  depth  of  the 
hold.  If  the  vessel  has  a  third  deck  then  the  height 
from  the  top  of  the  tonnage  deck  plank  to  the  under 
side  of  the  upper  deck  plank  is  accounted  as  the 
height  under  the  spar  deck. 

The  register  tonnage  of  a  vessel  is  her  internal 
cubical  capacity  in  tons  of  100  cubic  feet  each,  to  be 
ascertained  as  follows:  From  the  inside  of  the  inner 
plank  (average  thickness)  at  the  side  of  the  stem  to 
the  inside  of  the  plank  upon  the  stern  timbers  (aver- 
age thickness),  deducting  from  this  length  what  is 


COMMERCL\L    ARITHMETIC.  115 

due  to  the  rake  of  the  bow  in  the  thickness  of  the 
deck,  and  wliat  is  due  to  the  rake  of  the  stern  tim- 
ber in  the  thickness  of  the  deck,  and  also  wliat  is  due 
to  the  rake  of  the  stern  timber  in  one  third  of  the 
spring  of  the  beam. 

CLASSES.  I 

Class  1.  Vessels  of  which  the  tonnage  length  is' 
50  feet  or  under. 

2.  Over  60  feet,  and  not  exceeding  100  feet 
in  length. 

3.  Over  100  feet,  and  not  exceeding  150  feet 
in  length. 

4.  Over  150  feet,  and  not  exceeding  200  feet 
in  length. 

5.  Over  200  feet,  and  not  exceeding  250  feet 
in  length. 

6.  Ov^er  250  feet  in  length. 

If  there  is  a  break,  a  poop,  or  any  other  perma- 
nent closed  in  space  upon  the  upper  decks,  or  upon 
the  spar  deck,  available  for  cargo  or  stores,  or  for 
the  berthing  or  accommodation  of  passengers  or 
crew,  the  tonnage  of  «uch  space  is  computed. 

If  a  vessel  has  a  third  deck,  or  a  spar  deck,  the 
tonnage  of  the  space  between  it  and  the  tonnage 
deck  is  computed. 

In  computing  the  tonnage  of  open  vessels  the 
upper  edge  of  the  upper  strake  is  to  form  the  boun- 
dary line  of  measurement,  and  the  depth  should  be 
taken  from  an  athwart-ship  line,  extending  from  tlie 
upper  edge  of  said  strake  at  each  division  of  the 
length. 

The  register  of  a  vessel  expresses  the  nunil)cr  of 
decks,  the  tonnage  under  the  tonnage  deck,  tluit  of 
\he  between  decks  above  the  tonnage  deck,  also  that 


116  THE    MERCHANTS    AND    MECHANICS* 

of  tlic  poop  or  other  enclosed  space  above  the  deck, 
each  separately.  In  every  registered  United  States 
vessel  the  number  denoting  the  total  registered  ton- 
nage ninst  be  deeply  carved,  or  otherwise  perma- 
nently marked  upon  her  mnin  beam,  and  shall  be  so 
continued;  and  if  at  any  time  it  cease  to  be  so  con- 
tinued, such  vessel  shall  no  longer  be  recognized  as 
la  registered  United  States  vessel. 

RECAPITULATION    OF    MEASUREMENTS. 

Ilegistered  Length. — Length  at  the  middle  of  the 
second  deck  from  below  in  vessels  of  two  or  more 
decks,  and  in  all  other  vessels  of  the  upper  deck, 
measured  from  the  fore  part  of  the  outer  planking 
upon  the  side  of  the  stem  to  the  after  part  of  the 
main  stern  post  of  single  screw  propeller  steamers, 
and  to  the  after  part  of  the  rudder  post  of  other  ves- 
bc;ls,  measured  upon  the  top  of  the  tonnage  deck. 

Tonnage  Length. — Length  at  upper  side  of  ton- 
nage deck  beams  from  inside  of  the  inboard  plank, 
{it  its  average  thickness  at  the  side  of  the  stem,  to 
the  inside  of  the  plank  upon  the  stern  timbers,  at  its 
average  thickness,  deducting  from  this  length  that 
which  is  due  to  the  rake  of  the»bow  in  the  thickness 
of  tlie  deck,  and  of  the  stern  timber  in  the  thickness 
of  tlie  deck,  and  one  third  the  spring  of  the  beam. 

Breadth  of  Beam. — At  the  broadest  part  of  the 
outside  of  the  vessel. 

Depth  cf  Hold., — Height  measured  from  the  under 
side  of  tonnage  deck  plank  amidships,  from  a  point 
ut  a  distance  of  one  third  the  spring  of  the  beam 
to  the  ceiling  of  the  hold  at  its  average  thickness. 

Height  under  Spar  Deck. — The  mean  height  ^vom 
top  of  tonnage  deck  plank  to  the  under  side  of  t\\Q 
upper  deck  plank. 


COMMERCIAL    ARITHMETIC.  lit 

Open  Vessels. — The  upper  edge  of  the  upper  strake 
is  to  be  the  boundary  line  of  measurement  of  length, 
and  th  3  depth  is  to  be  measured  from  a  line  running 
athwait-ships  from  the  upper  edge  of  the  upper 
strake  at  each  division  of  the  length. 

By  ati  Act  of  Congress  of  28th  of  February,  1865, 
the  preceding  rule  of  admeasurement  was  amended 
as  folio  Wo:  No  part  of  any  ship  or  vessel  shall  be 
admeasured  or  registered  for  tonnage  that  is  used 
for  cabins  or  state-rooms,  and  constructed  entirely 
above  the  tirst  deck,  which  is  not  a  deck  to  the  hull. 

carpenter's    alEASUREMENT. FOR    A   SINGLE  DECK  VESSEL. 

Rule. — Multiply  the  length  of  keel,  the  breadth  of 
beam,  and  the  depth  of  hold  together,  and  divide 
by  a5. 

FOR    A    DOUBLE    DECK    VESSEL. 

Rule. — Multiply  as  above,  taking  half  the  breadth 
of  beam  for  the  depth  of  the  hold,  and  divide  by  95. 

BRITISH    MEASUREMENT. 

Divide  the  length  of  the  upper  deck  between  the 
after  part  of  the  stem  and  the  fore  part  of  the  stern 
post  into  6  equal  parts,  and  note  the  foremost,  middle 
and  aftermost  points  of  division.  Measure  the  depths 
at  t'lese  three  points  in  feet  and  tenths  of  a  foot,  also 
the  depths  from  the  under  side  of  the  upper  deck  to 
the  ceiling  of  the  limber  strake,  or,  in  case  ofa  break 
in  the  upper  deck,  from  a  line  stretched  in  continua- 
tion of  the  deck.  For  the  breadths  divide  each 
depth  into  5  equal  parts,  and  measure  the  inside 
breadths  at  the  following  points,  viz.,  at  .2  and  .8 
from  the  upper  deck  of  the  foremost  and  aftermost 
depths,  and  at  A  and  .8  from  the  upper  deck  of  the 
midsliip  depth.  Take  the  length  at  half  the  miJ- 
fthlp  dopth  from  the  after  part  of  the  stem  to  the 


118  THE    MERCHANTS    AND    MECHANICS^ 

fore  part  of  the  stern  post.  Then,  to  twice  the  mid- 
ship depth  add  the  foremost  and  aftermost  depthsi 
for  the  sum  of  the  depths,  and  add  together  the  fore- 
most upper  and  lower  breadths — 3  times  the  iippei 
breadths  witli  the  lower  breadth  at  the  midship,  and 
the  upper  and  twice  the  lower  breadth  at  the  after 
division  for  the  sum  of  the  breadths. 

Multiply  together  the  sum  of  the  depths,  the  sura 
of  the  breadths  and  length,  and  divide  the  product 
by  3500,  which  will  give  the  number  of  tons,  or  reg- 
ister. 

If  the  vessel  has  a  poop  or  half  deck,  or  a  break  in 
the  upper  deck,  measure  the  inside  mean  length, 
breadth  and  height  of  such  part  thereof  as  may  be 
included  within  the  bulkhead;  multiply  these  three 
measurements  together  and  divide  the  product  by 
92.4.  The  quotient  will  be  the  number  of  tons  to  be 
added  to  the  result  as  above  ascertained. 

For  Open  Vessels. — The  depths  are  to  be  taken 
from  the  upper  edge  of  the  upper  strake. 

For  Steam  Vessels. — The  tonnage  due  to  th^  en- 
gine room  is  deducted  from  the  total  tonnage  com- 
puted by  the  above  rule. 

To  determine  this  measure  the  inside  length  of 
the  engine  room  from  the  foremost  to  the  aftermost 
bulkhead,  then  multiply  this  length  by  the  mid- 
ship depth  of  the  vessel,  and  the  pro  luct  by  the  in- 
side midship  breadth  at  .4  of  the  depth  from  the 
deck,  and  divide  the  final  product  by  92.4. 

TABLE   OF  MULTIPLES. 

For  the  practical  convenience  of  those  who  have 
occasion  to  refer  to  mensuration,  we  have  arranged 
the  following  useful  table  of  multiples.     It  covers 


COMMERCIAL    ARITHMETIC.  119 

the  whole  ground  of  practical  geometry,  and  should 
be  studied  carefully  by  those  who  wish  to  be  skilled 
in  this  beautiful  branch  of  mathematics. 

Diameter  of  a  circleX3.1416=circumference. 

Radius  of  a  circleXC.283185==circumference. 

Square  of  the  radius  of  a  circleX3.141G==area. 

Square  of  the  diameter  of  a  circleX0.t854=area. 

Square  of  the  circumference  of  a  circlcXO. 07958= 
area. 

Half  the  circumference  of  a  circle X^y  half  its  di- 
ameter=area. 

Circumference  of  a  circleX0-159155=radius. 

Square   root  of  the  area  of  a  circleX^-50419=: 
radius. 

Circumference  of  a  circleXO-SlSSl^diametet. 

Square  root  of  the  area  of  a  circleX112838=diam- 
eter. 

Diameter  of  a  circle X0-86=side  of  inscribed  equi. 
lateral  triangle. 

Diameter  of  a  circleX0.70U=side  of  an  inscribed 
square. 

Circumference  of  a  circleX9.225:^side  of  vn  in- 
scribed square. 

Circumference  of  a  circleX0.282=side  of  ai\  <;qual 
square. 

Diameter  of  a   circleX0.8862=side  of  ajj  equt>/ 
square. 

Base  of  a  triangleXby  J  the  altitiide=are^. 
Multiplying  both  diameters   an^   .1854  U/gethM 
■Marea  of  an  ellipse. 


120  THE    MERCHANTS    AND    MECHANICS' 

Bufface  of  a  sphereXby  ^  of  its  diameter=solidity. 

Circumference  of  a  sphereXby  its   diameter=snr- 

face. 

Square  of  the  diameter  of  a  sphereX3.1416=snr- 
facc. 

Square  of  the  circumference  of  a  sphereX0.318n 

=snrfacfe. 

Cube  of  the  diameter  of  a  sphercX0.5236=soliditj. 

Cube  of  the  radius  of  a  sphereX4.1888=solidity 

Cube  of  the  circumference  of  a  sphereX0-01638'i[ 
=solidity. 

Square  root  df  the  surface  of  a  sphercX0-56419= 
diameter. 

Square  root  of  the  surface  of  a  sphereXl •'772454 
=circumference. 

Cube  root  of  the  solidity  of  a  sphereXl-2407= 
diameter. 

Cube  root  of  the  solidity  of  a  sphereX3.8978=cir- 
cumference. 

Radius  of  a  sphereXl-1547=side  of  inscribed  cube. 

Square  root  of  (|  of  the  square  of)  the  diameter  of 
a  sphere=side  of  inscribed  cube. 

Area  of  its  baseX^y  |  of  its  altitude=:solidity  of 
a  cone  or  pyramid,  whether  round,  square  or  trian- 
gular. 

Area  of  one  of  its  sidesX6=surface  of  a  cube. 

Altitude  of  rapezoidXIi  ^^^^  sum  of  its  parallel 
sides=-area. 


COMMERCIAL  ARITHMETia  121 


\AroOD,    TIMBER,   etc. 

Selection  of  Standing  Trees. —  Wood  grown  in  a 
moist  s»oil  is  lighter,  and  decays  sooner  than  that 
grown  in  dry,  sandy  soil. 

The  best  timber  is  that  grown  in  a  dark  soil  inter- 
mixed with  gravel.  Poplar,  cypress,  willow,  and  all 
others  which  grow  best  in  a  wet  soil,  are  exceptions. 

The  hardest  and  densest  woods,  and  the  least  sub- 
ject to  decay,  grow  in  warm  climates,  but  they  are 
more  liable  to  split  and  warp  in  seasoning. 
'V\Trees  grown  upon  plains  or  in  the  centre  of  forests 
are  less  dense  than  those  from  the  edge  of  a  forest, 
from  the  side  of  a  hill,  or  from  open  ground. 

Trees  (in  the  United  States)  should  be  selected  in 
the  latter  part  of  July  or  first  part  of  August — for 
at  this  season  the  leaves  of  the  sound,  healthy  trees 
are  fresh  and  green,  while  those  of  the  unsound  are 
beginning  to  turn  yellow.  A  sound,  healthy  tree  is 
recognized  by  its  top  branches  being  well  leaved, 
the  bark  even  and  of  a  uniform  color.  A  rounded 
top,  few  leaves,  some  of  them  turned  yellow,  a 
rougher  bark  than  common,  covered  with  parasitic 
plants,  and  with  streaks  or  spots  upon  it,  indicate  a 
tree  upon  the  decline.  The  decay  of  branches  and 
the  separation  of  bark  from  the  wood,  are  infallible 
indications  that  the  wood  is  impaired. 

Felling  Timber. — The  most  suitable  time  for  felling 
timber  is  in  midwinter  and  in  midsummer.  Recent 
experiments  indicate  the  latter  season,  and  in  the 
month  of  July. 

A  tree  should  be  allowed  to  attain  full  maturity 
before  l)eing  felled.  Oak  matures  at  75  to  100  years 
and  upward,  according  to  circumstances.  The  ags 
and  rate  of  growth  of  a  tree  are  indicated  by  the 


122         THE  MERCHANTS  AND  MECHANICS' 

number  and  width  of  the  rings  of  annual  incre^e 
which  are  exhibited  in  a  cross  section. 

A  tree  should  be  cut  as  near  the  ground  as  practi- 
cable, as  the  lower  part  furnishes  the  best  timber. 

Dressing  Timber. — As  soon  as  a  tree  is  felled  it 
should  be  stripped  of  its  bark,  raised  from  the 
ground,  the  sap  wood  taken  off  and  the  timber  re- 
duced to  its  required  dimensions. 

Inspection  of  Timber.— The  quality  of  wood  is  in 
some  degree  indicated  by  its  color,  which  should  be 
nearly  uniform  in  the  heart,  a  little  deeper  toward 
the  centre,  and  free  from  sudden  transitions  ofcolor. 
White  spots  indicate  decay.  The  sap  wood  is  known 
by  its  white  color;  it  is  next  to  the  bark,  and  very 
soon  rots. 

Defects  of  Timber. —  Wind-shakes  are  circular  cracks, 
separating  the  concentric  layers  of  wood  from  each 
other.     It  is  a  serious  defect. 

Splits,  checks  and  cracks,  extending  toward  the 
centre,  if  deep  and  strongly  marked,  render  the  tim- 
ber unfit  for  use,  unless  the  purpose  for  which  it  is 
intended  will  admit  of  its  being  split  through  them. 

Brushwood  is  generally  consequent  upon  the  de 
cline  of  the  tree  from  age.  The  wood  is  porous,  of  ^ 
reddish  color,  and  breaks  short  without  splinters. 

Belted  Timber  is  that  which  has  been  killed  beforr 
being  felled,  or  which  has  died  from  other  causes.  It 
is  objectionable. 

Knotty  Timber  is  that  containing  many  knots^ 
though  sound  ;  usually  of  stunted  growth. 

Twisted  Wood  is  when  the  grain  of  it  winds  spi- 
rally; it  is  unfit  for  long  pieces. 


COIIMEKCIAT,    ARITinrETIC.  123 

Dry  rot. — This  is  indicated  by  yellow  stains.  Elm 
and  beech  are  soon  affected  if  left  with  the  bark  on. 

Large  or  decayed  knots  injuriously  affect  the 
strength  of  timber. 

Seasoning  and  Preserving  Timber. 

Timber  freshly  cut  contains  about  3t  to  48  per 
cent,  of  liquids.  By  exposure  to  the  air  in  seasoning 
one  year  it  loses  from  IT  to  25  per  cent.;  and,  when 
seasoned,  it  yet  retains  from  10  to  15  per  cent. 

Timber  of  large  dimensions  is  Improved  and  ren- 
dered less  liable  to  warp  and  crack  in  being  seasoned 
by  immersion  in  water  for  some  weeks. 

For  the  purpose  of  seasoning  tiniber  should  bo 
piled  under  shelter  and  be  kept  dry.  It  should  have 
a  free  circulation  of  air  about  it,  without  being  ex- 
posed to  strong  currents.  The  bottom  pieces  should 
be  placed  upon  skids,  which  should  be  free  from 
decay,  rai.sed  not  less  than  two  feet  from  the  ground: 
a  space  of  an  inch  should  intervene  betv/een  the 
pieces  of  the  same  horizontal  layers,  and  slats  or 
piHng  strips  placed  between  each  layer,  one  near 
each  end  of  the  pile  and  others  at  short  distances,  in 
order  to  keep  the  timber  from  winding.  These  strips 
should  be  one  over  the  other — and,  in  large  piles, 
should  not  be  less  than  one  inch  thick.  Light  tim- 
ber may  be  piled  in  the  upper  portion  of  tlie  shelter, 
heavy  timber  upon  the  ground  floor.  Each  pile 
should  contain  but  one  description  of  timber.  The 
piles  siioiiid  be  at  least  two  and  a  half  feet  apart. 

Timber  should  bo  replied  at  intervals,  and  all 
pieces  indicating  decay  should  be  removed,  to  pre- 
vent their  affecting  those  which  are  still  sound. 

Timber  houses  ore  best  provided  with  blinds, 
which  keep  out  rain  nnd  pnow,  l)Mt  which  can  bo 


124  THE    MERCHANTS    AND    MECHANICS' 

turned  to  admit  air  in  fine  weather;  and  they  should 
be  kept  entirely  free  from  any  pieces  of  decayed 
wood. 

The  gradual  mode  of  seasoning  is  the  most  favora- 
ble to  the  strength  and  durability  of  timber  ;  but 
various  methods  have  been  proposed  for  hastening 
the  process.  For  this  purpose  steaming  timber  has 
been  applied  with  success,  and  the  results  of  experi- 
ments of  various  processes  of  saturating  timber  with 
a  solution  of  corrosive  sublimate  and  antiseptic  fluids 
are  very  satisfactory.  This  process  hardens  and  sea- 
sons wood,  at  the  same  time  that  it  secures  it  from 
dry  rot  and  from  the  attacks  of  worms.  Kiln  drying 
is  serviceable  only  for  boards  and  pieces  of  small 
dimensions,  and  is  apt  to  cause  cracks  and  to  impair 
the  strength  of  wood,  unless  performed  very  slowly. 
Charring  or  painting  is  highly  injurious  to  any  but 
seasoned  timber,  as  it  effectually  prevents  the  dry- 
ing of  the  inner  part  of  the  wood,  in  consequence  of 
which  fermentation  and  decay  soon  take  place. 

Timber  piled  in  badly  ventilated  sheds  is  apt  to 
be  attacked  with  the  common  rot.  The  first  out- 
ward indications  are  yellow  spots  upon  the  ends. of 
the  pieces,  and  a  yellowish  dust  in  the  checks  and 
cracks,  particularly  where  the  pieces  rest  upon  the 
piling  strips. 

Timber  requires  from  two  to  eight  years  to  be 
seasoned  thoroughly,  according  to  its  dimensions.  It 
should  be  worked  as  soon  as  it  is  thoroughly  dry,  for 
it  deteriorates  after  that  time. 

Oak  timber  loses  one  fifth  of  its  weight  in  season- 
ing, and  about  one  third  of  its  weight  in  becoming 
perfectly  dry.  Seasoning  is  the  extraction  or  dissi- 
pation of  the  vegetable  juices  and  moisture,  or  the 
solidification  of  the  albumen.  When  wood  is  exposed 
to  currents  of  air  at  a  high  temj^rature  the  moisture 


COMMF.IICIA!.    ARITHMETIC.  125 

evaporates  too  rapidly  and  the  wood  cracks,  and 
when  the  temperature  is  high  and  sap  remains  it 
ferments,  and  dry  rot  ensues. 

Timber  is  subject  to  common  rot  or  dry  rot — the 
former  occasioned  by  alternate  exposure  to  moisture 
and  dryness.  The  progress  of  this  decay  is  from  the 
exterior;  hence,  the  covering  of  the  surface  with 
paint,  tar,  etc.,  is  a  preservative. 

Painting  and  charring  green  timber  hastens  its 
decay. 

Dry  or  sap  rot  is  inherent  in  timber,  and  it  is 
occasioned  by  the  putrefaction  of  the  vegetable  albu- 
men. Sap  wood  contains  a  large  proportion  of  fer- 
mentable elements.  Insects  attack  wood  for  the 
sugar  or  gum -contained  in  it,  and  fungi  subsist  upon 
the  albumen  of  wood  ;  hence,  to  arrest  dry  rot  the 
albumen  must  be  either  extracted  or  solidified. 

In  the  seasoning  of  timber  naturally  there  is  re- 
quired a  period  of  from  two  to  four  years.  Immer- 
sion in  water  facilitates  seasoning  by  solving  the  sap. 

The  most  effective  method  of  preserving  timber  is 
that  of  expelling  or  exhausting  its  fluids,  solidifying 
its  albumen  and  introducing  an  antiseptic  liquid. 

The  strength  of  impregnated  timber  is  not  re- 
duced, and  its  resilience  is  improved. 

In  desiccating  timber,  by  expelling  its  fluids  by 
heat  and  air,  its  strength  is  increased  fully  15  per 
cent. 

In  coating  unseasoned  timber  with  creosote,  tar, 
etc.,  it  is  also  preserved  from  the  attack  of  worms. 
Jarrow  wood,  from  Australia,  is  not  subjected  to 
their  attack. 

The  condition  of  timber,  as  to  its  soundness  or 
decay,  is  readily  recognized  when  struck  a  quick 
blow. 

Timber  that  has  been  for  a  long  time  immersed  in 


126  THE    MERCHANTS    AND    MECHANICS* 

water,  when  brought  into  the  air  and  dried,  becomes 
brashy  and  useless. 

When  trees  are  barked  in  the  spring  they  should 
not  be  lelled  until  the  foliage  is  dead. 

Timber  cannot  be  seasoned  by  either  smoking  or 
charring;  but  when  it  is  to  be  used  in  locations 
where  it  is  exposed  to  worms  or  to  produce  fungi,  it 
is  proper  to  smoke  or  char  it. 

Timber  may  be  partially  seasoned  by  beting  boiled 
or  steamed 

Impregnation  of  Wood. 

The  several  processes  are  as  follows  : 

Kyan,  1832. — Saturated  with  corrosive  sublimate. 
Solution,  1  lb.  of  chloride  of  mercury  to  4  gallons 
of  water 

Burnett,  1838. — Impregnation  with  chloride  of  zinc, 
by  submitting  the  wood  endwise  to  a  pressure  of  150 
lbs.  per  square  inch.  Solution,  1  lb.  of  the  chloride 
to  10  gallons  of  water. 

Boucheri. — Impregnation,  by  submitting  the  wood 
endwise  to  a  pressure  of  about  15  lbs.  per  square 
inch.  Solution,  1  lb.  of  sulphate  of  copper  to  12J 
gallons  of  water. 

^e^A.eZ.— Impregnation,  by  submitting  the  wood  end- 
Vrise  to  a  pressure  of  150  to  200  lbs.  per  square  inch, 
with  oil  of  creosote  mixed  with  bituminous  matter. 

Louis  S.  Bobbins,  1865. — Aqueous  vapor  dissipated 
by  the  wood  being  heated  in  a  chamber,  the  albu- 
men solidified  then  submitted  to  the  vapor  of  coal 
tar,  resin,  or  bituminous  oils,  which,  being  at  a  tem- 
perature not  less  than  325°,  readily  takes  the  place 
of  the  vapor  expelled  by  a  temperature  of  212°. 


COMMERCIAL    ARITHMETIC. 


121 


Fluids  will  pass  with  the  grain  of  wood  with -great 
/acility,  but  will  not  enter  it  except  to  a  very  limited 
extent  when  applied  externally. 

Absorption  of  Preserving  Solution  by  differ- 
ent Woods  for  a  per^iod  of  seven  days. 


AVERAGE  POUNDS  PER  CUBIC  FOOT. 

Black  Oak 3,6  I  Hen>lock 2.6  I  Rock  Oak 3.9 

Chestnut 3.    |  Red  Oak 3.9  j  White  Oak 3.1 


PROPORTION   OS"   YTATER   IX    VARIOUS   WOODS. 


Alder  {Betula  alnus) 41.6 

Ash  {Fraxinus excelsior) 28.7 

Birch  {Betula  alba) 30.8 

Elm  (  Vlmus  camperi-^i) 44.5 

Horse  Chestnut  {^Esculus  hip- 

pocast) 88.2 

Iatc^h  Pinm  larix) 48.6 

Mountain  Ash  (Sorbus  aucupa- 

ria)  28.3 

Oak  {Querau  robur) 34.7 

COMPARATIVE   RESILIENCE    OF   TIMBER. 

Asli l.OOjChestnnt 73|Larch ^iSpruce 

Beech 86  Elm 54  Oak 63  Teak 

Cedar 66  Fir .41  Pitch  Piue 57 1  Yellow  Pine. 


Pine  (Pinus  Sylvestris  L.) 

Red  Beech  (Fagus  sylvatica) . . 
Red  V'lnQ  (Pinus picea  dur) .. . 
Sycamore  (Acer  pseudo-plata- 

nus) 

White  Oak  ( Ouercus  alba) .... 
White  Pine  (Pimis  dbies  dur) . . 
White  Poplar  (Populus  alba).. 
Willow  {Salix  caprea) 


39.7 
39.7 
45.2 

27. 

36.2 

37.1 

50.6 

26. 


.64 


.64 


Weight  and  Strength  of  Oak  and  Yellow 
Pine. 

WEIGHT    OF   A    CUBIC    FOOT. 


Age. 

White  Oak,  Va. 

Yellow  Pine,   Va. 

Live  Oak 

Hound. 

Sqoare. 

nonnd. 

Square. 

Green 

64.7 
53.6 
46. 

67. 7 
53.5 
49.9 

47.8 

39.8 
34.8 

39.2 
34.2 
83.5 

78  7 

One  Year  '.  

Two  Years 

66.7 

In  England  timber  sawed  into  boards  is  classed 
^s  folio w.s : 

Gi  to  t  inches  in  width,  Battens;  8 J  to  10  inches, 
Veals;  and  11  to  12  inches,  Planks. 


X28  THE    MERCHANTS    AND    MECHANICS*  , 

In  a  perfectly  dry  atmosphere  the  durability  of 
woods  is  almost  unlimited.  Rafters  of  roofs  are 
known  to  have  existed  1,000  years,  and  piles  sub- 
merged in  fresh  Avater  have  been  found  perfectly 
sound  800  years  from  the  period  of  their  being 
driven. 

Distillation. — From  a  single  cord  of  pitch  pine,  dis- 
tilled by  chemical  apparatus,  the  following  substances, 
and  in  the  quantities  stated,  have  been  obtained: 


Charcoal 50  bush. 

Illuminating  Gas... about  1000  cu.  ft. 
Illuminating  Oil  •J-ud  Tar  50  galls- 
Pitch  or  Resin I>^bbl3. 


Pyroligneoua  Acid 100  galls. 

Spirits  of  Turpentine ....    20     '  • 

Tar ,      1  bbl. 

Wood  Spirit 5  galls. 


DECREASE   IN   DIMENSIONS   OF  TIMBER   BY    SEASONING. 

Woortg.  Ing.  Ing.  WoodB.  Ipb.  Ins. 


Cedar,  Canada 14  to   13  J^ 

Elm 11   to   mx 

Oak.  English 13   to   11 V^ 

Pitch  Pine,  North..l0xl0  to  9Xx9X 


Pitch  Pine,  Sonth lfe%  to  18)^ 

Spruce 8>i  to    8% 

White  Pine,  American.  12      to  11% 
Yellow  Pine,  North ...  18     to  17% 


The  weight  of  a  beam  of  English  oak,  when  wet, 
was  reduced  by  seasoning  from  9t2.25  to  630.5  lbs. 

IRON. 
The  foreign  substances  which  iron  contains  modify 
its  essential  properties.  Carbon  adds  to  its  hard- 
ness, but  destroys  some  of  its  qualities,  and  produces 
cast  iron  or  steel  according  to  the  proportion  it  con- 
tains. Sulphur  renders  it  fusible,  difficult  to  weld, 
and  brittle  when  heated,  or  "  hot  short."  Phospho- 
rus renders  it  "  cold  short,"  but  may  be  present  in 
the  proportion  of  y^^  to  nfcr  without  a^ecting  in- 
juriously its  tenacity.  Antimony,  arsenic  ^nd  cop- 
per have  the  same  effect  as  sulphur,  the  last  in  a 
greater  degree. 

Cast  Iron. 

The  process  of  making  cast  iron  depends  wrch 
upon  the  description  of  fuel  used — whether  charcoal, 
coke,  bituminous  or  anthracite  coals.    A  larger  vi^vl«i 


COmiERGlAL    ARITHMETIC.  j  125 

from  the  same  furnace,  and  a  great  economy  in 
fuel  are  effected  by  the  use  of  a  hot  blast.  The  greater 
heat  thus  produced  causes  the  iron  to  combine  with 
a  larger  percentage  of  foreign  substances. 

Cast  iron,  for  purposes  requiring  great  strength, 
should  be  smelted  with  a  cold  blast.  Pig  iron,  ac- 
cording to  the  proportion  of  carbon  which  it  con- 
tains, is  divided  into  foundry  iron  and  forge  iron — 
the  latter  adapted  only  to  conversion  into  malleable 
iron,  while  the  former,  containing  the  largest  pro- 
portion of  carbon,  can  be  used  either  for  castings  or 
bars. 

There  are  many  varieties  of  cast  iron,  differing  by 
almost  insensible  shades;  the  two  principal  divisions 
are  gray  and  ivhite,  so  termed  from  the  color  of  their 
fracture.     Their  properties  are  very  different. 

Gray  iron  is  softer  and  less  brittle  than  white  iron; 
it  is  in  a  slight  degree  malleable  and  flexible,  and  is 
not  sonorous;  it  can  be  easily  drilled  or  turned  in  a 
lathe,  and  does  not  resist  the  file.  It  has  a  brilliant 
fracture,  of  a  gray  (or  sometimes  a  bluish-gray)  color. 
The  color  is  lighter  as  the  grain  becomes  closer,  and 
its  hardness  increases  at  the  same  time.  It  melts  at 
a  lower  heat  than  white  iron,  and  preserves  its  flu- 
idity longer.  The  color  of  the  fluid  metal  is  red, 
and  deeper  in  proportion  as  the  heat  is  lower;  it 
does  not  adhere  to  the  ladle;  it  fills  the  moulds  well, 
contracts  less,  and  contains  fewer  cavities  than  white 
iron;  the  edges  of  its  castings  are  sharp  and  the  sur- 
faces smooth  and  convex.  A  medium  sized  grain, 
bright  gray  color,  fracture  sharp  to  the  touch,  and  a 
close,  compact  texture,  indicate  a  good  quality  of 
iron.  A  grain  either  very  large  or  very  small,  a 
dull,  earthy  aspect,  loose  texture,  dissimilar  crystals 
mixed  together,  indicate  an  inferior  quality. 

Gray  iron  is  used  for  machinery  and   ordnance 


130  THE    MERCHANTS    AND    MECHANICS' 

purposes,  where  the  pieces  are  to  be  bored  or  fitted. 
Its  tenacity  and  specific  gravity  are  diminished  by 
annealing.     Its  mean  specific  gravity  is  1.2. 

White  iron  is  very  brittle  and  sonorous;  it  resists 
the  file  and  the  chisel,  and  is  susceptible  of  high 
polish.  The  surface  of  its  castings  is  concave;  the 
fracture  presents  a  silvery  appearance,  generally  fine 
grained  and  compact,  sometimes  radiating  or  lamel- 
lar. When  melted  it  is  white  and  throws  off  a  great 
number  of  sparks,  and  its  qualities  are  the  reverse  of 
those  of  gray  iron;  it  is,  therefore,  unsuitable  for 
machinery  purposes.  Its  tenacity  is  increased  and 
its  specific  gravity  diminished  by  annealing.  Its 
mean  specific  gravity  is  1.5, 

Mottled  iron  is  a  mixture  of  white  and  gray;  it  has 
a  spotted  appearance;  it  flows  well  and  with  few 
sparks;  its  castings  have  a  plain  surface  with  edges 
slightly  rounded  ;  it  is  suitable  for  shot,  shell,  etc. 

A  fine  mottled  iron  is  the  only  kind  suitable  for 
castings  which  require  great  strength,  such  as  beam 
centres,  cylinders  and  cannon.  The  kind  of  mottle 
will  depend  much  upon  the  size  of  the  casting. 

Besides  these  general  divisions  the  dift'erent  varie- 
ties of  pig  iron  are  more  particularly  distinguished 
by  numbers,  according  to  their  relative  hardness. 

No.  1  is  the  softest  iron,  possessing  in  the  highest 
degree  the  qualities  belonging  to  gray  iron;  it  has 
not  much  strength,  but  on  account  of  its  fluidity 
when  melted,  and  of  its  mixing  advantageously  with 
old  or  scrap  iron,  and  with  the  harder  kinds  of  cast 
iron,  it  is  of  great  use  to  the  founder,  and  commands 
the  highest  price. 

No.  2  is  harder,  closer  grained  and  stronger  than 
No.  I;  it  has  a  gray  color  and  considerable  lustre. 
It  is  the  character  of  iron  most  suitable  for  shot  and 
shell. 


COMMERCIAL   ARITHMETIC.  131 

No.  3  rs  ^till  harder  than  No.  iJ  ;  its  color  is  gray, 
but  inclining:  to  white;  it  has  considerable  strength, 
but  it  is  principally  used  for  mixing  with  other  kinds 
of  iron. 

No.  4  is  bright  iron;  No.  5  mottled,  and  No.  6 
n^hite,  which  is  unfit  for  general  use  by  itself. 

The  quality  of  these  various  descriptions  depend 
jipon  the  proportion  of  carbon,  and  upon  the  state 
xn  which  it  exists  in  the  metal.  In  the  darker  kinds 
of  iron,  where  the  proportion  is  sometimes  1  per 
cent.,  it  exists  partly  in  the  state  of  graphite  or 
plumbago,  which  makes  tlie  iron  soft.  In  white  iron 
the  carbon  is  thoroughl}^  combined  with  the  metal, 
as  in  steel. 

Cast  iron  frequently  retains  a  portion  of  foreign 
ingredients  from  the  ore,  such  as  earths  or  oxides  of 
other  metals,  and  sometimes  sulphur  and  phospho- 
rus, which  are  all  injurious  to  its  quahty.  Sulphur 
hardens  the  iron,  and,  unless  in  a  very  small  propor- 
tion, destroys  its  tenacity. 

These  foreign  substances,  and  also  a  portion  of  the 
carbon,  are  separated  by  melting  the  iron  in  contact 
witli  air,  and  soft  iron  is  thus  rendered  harder  and 
stronger.  The  effect  of  remelting  varies  with  the 
nature  of  the  iron  and  the  character  of  the  ore  from 
which  it  has  been  extracted — that  from  the  hard 
ores,  such  as  the  magnetic  oxides,  undergoes  less 
alteration  than  that  from  the  hematites,  the  latter 
being  sometimes  changed  from  No.  1  to  white  by  a 
single  remelting  in  an  air  furnace. 

The  color  and  texture  of  cast  iron  depend  greatly 
upon  the  volume  of  the  casting  and  the  rapidity  of 
its  cooling — a  small  casting,  which  cools  quickly,  is 
almost  always  white,  and  the  surface  of  large  cast- 
ings partakes  more  of  the  qualities  of  white  metal 
than  the  interior. 


132  THE    MERCHANTS    AND    MECHANICS^ 

All  cast  iron  expands  at  the  moment  of  becoming 
solid  and  contracts  in  cooling.  Gray  iron  expands 
more  and  contracts  less  than  other  iron.  The  con- 
traction is  about  Y^-g-  for  gray  and  strongly  mottled 
iron,  or  |  of  an  inch  per  foot. 

Remelting  iron  improves  its  tenacity;  thus,  a  mean 
of  14  cases  for  two  fusions  gave,  for  1st  fusion,  a 
tenacity  of  29,284  lbs.;  for  2d  fusion,  33,t90  lbs. 
And  two  cases,  for  1st  fusion,  15.129  lbs.;  for  2d  fu- 
sion, 35,786  lbs. 

Wrought  Iron. 

Wrought  iron  is  made  from  the  pig  iron  in  a 
bloomery  fire  or  in  a  puddling  furnace — generally  in 
the  latter.  The  process  consists  in  melting  it  and 
keeping  it  exposed  to  a  great  heat,  constantly  stir- 
ring the  mass,  bringing  every  part  of  it  under  the 
action  of  the  flame  until  it  loses  its  remaining  car- 
bon, when  it  becomes  malleable  iron.  AVhen,  how- 
ever, it  is  desired  to  obtain  iron  of  the  best  quality, 
the  pig  iron  should  be  refined. 

Refining. — This  operation  deprives  the  iron  of  a 
considerable  portion  of  its  carbon ;  it  is  effected  in  a 
blast  furnace,  where  the  iron  is  melted  by  means  of 
charcoal  or  coke,  and  exposed  for  some  time  to  the 
action  of  a  great  heat;  the  metal  is  then  run  into  a 
cast  iron  mould,  by  which  it  is  formed  into  a  large 
broad  plate.  As  soon  as  the  surface  of  the  plate  is 
chilled  cold  water  is  poured  on  to  render  it  brittle. 

The  bloomery  resembles  a  large  forge  fire,  where 
charcoal  and  a  strong  blast  are  used;  and  the  refined 
metal  or  the  pig  iron,  after  being  broken  into  pieces 
of  the  proper  size,  is  placed  before  the  blast,  directly 
in  contact  with  charcoal;  as  the  metal  fuses  it  falls 
into  a  cavity  left  for  that  purpose  below  the  blast, 
where  the  bloomer  works  it  into  the  shape  of  a  ball. 


COMMERCIAL    ARITHMETIC.  133 

which  he  places  again  before  the  blast,  with  fresh 
charcoal;  this  operation  is  generally  again  repeated, 
when  the  ball  is  ready  for  the  shingler. 

The  puddling  furnace  is  a  reverberatory  furnace, 
where  the  flame  of  bituminous  coal  is  brought  to  act 
directly  upon  the  metal.  The  metal  is  first  melted, 
the  puddler  then  stirs  it,  exposing  each  portion  in 
turn  to  the  action  of  the  flame,  and  continues  this  as 
long  as  he  is  able  to  work  it.  When  it  has  lost  its 
fluidity  he  forms  it  into  balls  weighing  from  80  to 
100  pounds,  which  are  next  passed  to  the  shingler. 

Shingling  is  performed  in  a  strong  squeezer,  or 
under  tiie  trip-hammer.  Its  object  is  to  press  out  as 
perfectly  as  practicable  the  liquid  cinder  which  the 
ball  still  contains;  it  also  forms  the  ball  into  shape 
for  the  puddle  rolls.  A  heavy  hammer,  weighing 
from  six  to  seven  tons,  effects  this  object  most  tho- 
roughly, but  not  so  cheaply  as  the  squeezer.  The 
ball  receives  from  fifteen  to  twenty  blows  of  a  ham- 
mer, being  turned  from  tiijie  to  time,  as  required;  it 
is  now  termed  a  bloom,  and  is  rea^  ^  to  be  rolled  or 
hammered;  or  the  ball  is  passed  Oi /"e  through  the 
squeezer  and  is  still  hot  enough  to  be  p^sed  through 
the  puddle  rolls. 

Puddle  Rolls. — By  passing  the  bloom  through  dif- 
ferent grooves  in  these  rolls  it  is  reduced  to  a  rough 
bar,  from  three  to  four  feet  in  length — its  name  con- 
veying an  idea  of  its  condition,  which  is  rough  and 
imperfect. 

Piling. — To  prepare  rough  bars  for  this  operation 
they  are  cut  by  a  pair  of  shears  into  such  lengths  as 
arc  best  adapted  to  the  size  of  the  finished  bar  re- 
quired; the  sheared  bars  are  then  piled  one  over  the 
other,  according  to  the  volume  required,  when  the 
pile  is  ready  for  balling. 


134  THK    .MKRCHAXTS    AND    MECTIAXICS' 

Balling. — This  operation  is  performed  in  the  ball- 
ing furnace,  which  is  similar  to  the  puddling  furnace, 
except  that  its  bottom  or  hearth  is  made  up,  from 
time  to  time,  with  sand;  it  is  used  to  give  a  welding 
heat  to  the  piles  to  prepare  them  for  rolling. 

Finishing  Rolls. — The  balls  are  passed  succes- 
sively between  rollers  of  various  forms  and  dimen- 
sions, according  to  the  shape  of  the  finished  bar 
required. 

The  quality  of  the  iron  depends  upon  the  descrip- 
tion of  pig  iron  used,  the  skill  of  the  puddler,  and  the 
absence  of  deleterious  substances  in  the  furnace. 

The  strongest  cast  irons  do  not  produce  the  strong- 
est malleable  iron. 

For  many  purposes — such  as  sheets  for  tinning, 
best  boiler  plates,  and  bars  for  converting  into  steel 
— charcoal  iron  is  used  exclusively;  and,  generally, 
this  kind  of  iron  is  to  be  relied  upon,  for  strength 
and  toughness,  with  greater  confidence  than  any 
other,  though  iron  of  superior  quality  is  made  from 
pigs  made  with  other  fuel  and  with  a  hot  blast. 
Iron  for  gun  barrels  has  been  lately  made  from  an- 
thracite hot  blast  pigs. 

Iron  is  improved  in  quality  by  judicious  working, 
reheating  it,  and  hammering  or  rolling.  Other  things 
being  equal,  the  best  iron  is  that  which  has  been 
wrought  the  most. 

STEEL. 

Steel  is  a  compound  of  iron  and  carbon,  in  which 
the  proportion  of  the  latter  is  from  1  to  5  per  cent., 
and  even  less  in  some  kinds.  Steel  is  distinguished 
from  iron  by  its  fine  grain,  and  by  the  action  of 
diluted  nitric  acid,  which  leaves  a  black  spot  upon 
steel,  and  upon  iron  a  spot  which  is  lighter  colored, 
in  proportion  to  the  carbon  it  contains. 


COMMERCIAL    ARITHMETIC.  135 

T\iere  are  many  varieties  of  steel,  the  principal  of 
which  are  : 

Natural  Sled,  obtained  by  reducing  rich  and  pure 
descriptions  of  iron  ore  with  cliarcoal,  and  refining 
the  cast  iron  so  a.i  to  deprive  it  of  a  sufficient  por- 
tion of  carbon  to  bring  it  to  a  malleable  state.  It  is 
used  for  files  and  other  tools. 

Indian  Steel,  termed  luoofz,  is  said  to-  be  a  natural 
stocl,  containing  a  small  portion  of  other  metals. 

Blistered  Steel,  or  steel  of  cementation,  is  prepared 
by  the  direct  combination  of  iron  and  carbon.  For 
this  purpose  the  iron,  in  bars,  is  put  in  layers,  alter- 
nating with  powJored  charcoal,  in  a  close  furnace, 
and  exposed  for  seven  or  eight  days  to  a  heat  of 
about  9000°,  and  then  put  to  cool  for  a  like  period. 
The  bars  on  being  taken  out  are  covered  with  blis- 
ters, have  acquired  a  brittle  quality,  and  exhibit  in 
the  fracture  a  uniform  crystalline  appearance.  The 
degree  of  carbonization  is  varied  according  to  the 
purposes  for  wiiicli  the  steel  is  intended,  and  the 
best  qualities  of  iron  (Russian  and  Swedish)  are 
used  for  the  finest  kinds  of  steel. 

Tilted  Steel  is  made  from  blistered  steel  moder- 
ately heated,  and  subjected  to  the  action  of  a  tilt 
hammer,  by  which  means  its  tenacity  and  density 
are  increased. 

Shear  Sled  is  made  from  blistered  or  natural  steel, 
refined  by  piling  thin  bars  into  fagots,  which  are 
brought  to  a  wielding  heat  in  a  reverberatory  furnace, 
and  hammered  or  rolled  again  into  bars.  This  opera- 
tion is  rej>eat  j;l  k^vcimI  times  to  produce  the  finest 
kinds  of  sheai*  si.cl,  which  are  distinguished  by  the 
names  of  "  half  sliear,"  "  single  shear"  and  "  double 
shear;"  or  steel  of  one,  two  or  three  marlca,  etc.,  ac- 
cording to  the  number  of  times  it  has  been  piled. 


136  THE    MERCHANTS    AND    MECHANICS' 

Cast  Steel  is  made  by  breaking  blistered  steel  into 
small  pieces  and  meltin<^  it  in  close  crucibles,  from 
which  it  is  poured  into  iron  moulds;  the  ingot  is 
then  reduced  to  a  bar  by  hammering  or  rolling. 
Cast  steel  is  the  best  kind  of  steel,  and  best  adapted 
for  most  purposes.  It  is  known  by  a  very  fme,  even 
and  close  grain,  and  a  silvery,  homogeneous  fracture; 
it  is  very  brittle  and  acquires  extreme  hardness,  but 
is  difficult  to  weld  without  the  use  of  a  flux.-  The 
other  kinds  of  steel  have  a  similar  appearance  to  cast 
steel,  but  the  grain  is  coarser  and  less  homogeneous; 
they  are  softer  and  less  brittle,  and  weld  more  read- 
ily. A  fibrous  or  lamellar  appearance  in  the  frac- 
ture indicates  an  imperfect  steel.  A  material  of 
great  toughness  and  elasticity,  as  well  as  hardness, 
is  made  by  forging  together  steel  and  iron,  forming 
the  celebrated  damasked  steel,  which  is  used  for 
sword  blades,  springs,  etc.,  the  damask  appearance 
of  which  is  produced  by  a  diluted  acid,  which  gives  a 
black  tint  to  the  steel  while  the  iron  remains  white. 

Various  fancy  steels,  or  alloys  of  steel  with  silver, 
platinum,  rhodium  and  aluminum,  have  been  made 
with  a  view  to  imitating  the  Damascus  steel,  wootz, 
etc.,  and  improving  the  fabrication  of  some  of  the 
finer  kinds  of  surgical  and  other  instruments. 

Properties  of  Steel. — After  being  tempered  it  is  not 
easily  broken;  it  welds  readily;  it  does  not  crack  or 
split;  it  bears  a  very  high  heat,  and  preserves  the 
capability  of  hardening  after  repeated  working. 

Hardening  and  Tempering. — Upon  these  opera- 
tions the  quality  of  manufactured  steel  in  a  great 
ileasure  depends. 

Hardening  is  effected  by  heating  the  steel  to  a 
cherry  red,  or  until  the  scales  of  oxide  are  loosened 
on  the  surface,  and  plunging  it  into  a  liquid,  or  i)la- 


COMMERCIAL    ARITHMETIC.  13^ 

cing  it  in  contact  with  some  cooling  substance.  The 
degree  of  hardness  depends  upon  the  heat  and  the 
rapidity  of  cooling.  Steel  is  thus  rendered  so  hard 
as  to  resist  the  hardest  files,  and  it  becomes  at  the 
same  time  extremely  brittle.  The  degree  of  heat 
and  the  temperature  and  nature  of  the  cooling  me- 
dium miiet  be  chosen  with  reference  to  the  quality 
of  the  steel  and  the  purpose  for  which  it  is  intended. 
Cold  water  gives  a  greater  hardness  than  oils  or 
other  fatty  substances,  sand,  wet  iron  scales  or  cin- 
ders, but  an  inferior  degree  of  hardness  to  that  given 
by  acids.  Oil,  tallow,  etc.,  prevent  the  cracks  which 
are  caused  by  too  rapid  cooling.  The  lower  the  heat 
at  which  the  steel  becomes  hard  the  better. 

Tempering. — Steel,  in  its  hardest  state,  being  too 
brittle  for  most  purposes,  the  requisite  strength  and 
elasticity  are  obtained  by  tempering — or  letting  down 
the  temper,  as  it  is  termed — which  is  performed  by 
heating  the  hardened  steel  to  a  certain  degree  and 
cooling  it  quickly.  The  requisite  heat  is  usually 
ascertained  by  the  color  which  the  surface  of  the 
steel  assumes  from  the  film  of  oxide  thus  formed. 
The  degrees  of  heat  to  which  these  several  colors 
correspond  are  as  follows  : 

At  430**,  a  very  faint  yellow  j  Suitable  for  hard  instruments  ;  as  hammer 
At 450*,  a  pale  straw  color.  |     faces,  drills,  etc. 

At  470®  a  full  vellow  ( ^°'"  instraments  requiring  hard  edges  with> 

A t  2qoo  o  w  Jn  «^i^; \     out  elasticity ;  as  shears,  Bcissors,  turn- 

At  490«,  a  brown  color j     j^g  ^^^j^^  ^^'^  ' 

^ISte'  ^'"°^'*'  '"'*'  ^"^'^  f  ^o'"  t«o'»  fo"-  cutting  wood  and  soft  metals ; 

At  538"   Durole* \     ^'^ch  as  plane  irons,  knives,  etc. 

AtViOo    Hnrk  Wn«  (For  tools  requirinsc  strong  edges  without 

At  SW*'  full  blir extreme  hardness  ;  as  cold  chisels,  ftxes, 

'  (     cutlery,  etc. 

At  600*,  grayish  blue,  verg-  j  For  spring  temper,  which  will  bend  before 
ing  on  black j     breaking ;  as  saws,  sword  blades,  etc. 

If  the  steel  is  heated  higher  than  this  the  effect 
of  the  hardening  process  is  destroyed. 


138  THE    MERCHANTS    AXD    MECHANICS' 


Case-hardening. 

This  operation  consists  in  converting  the  surface 
of  wrought  iron  into  steel  by  cementation,  for  the 
purpose  of  adapting  it  to  receive  a  polish  or  to  bear 
friction,  etc.  This  is  effected  by  heating  iron  to  a 
cherry  red  in  a  close  vessel,  in  contact  with  carbon- 
aceous materials,  and  then  plunging  it  into  cold 
water.  Bones,  leather,  hoofs,  and  horns  of  animals 
are  generally  used  for  this  purpose,  after  having 
been  burned  or  roasted  so  that  they  can  be  pulver- 
ized.    Soot  is  also  frequently  used. 

LIMES,    CEMENTS    AND    MORTARS. 
Limestones. 

The  calcination  of  marble,  or  any  pure  limestone, 
produces  lime  {quick-lime).  The  pure  limestones 
burn  white  and  give  the  richest  limes. 

The  finest  calcareous  minerals  are  the  rhombohe- 
dral  prisms  of  calcareous  spar,  the  transparent  dou- 
ble reflecting  Iceland  spar,  and  white  or  statuary 
marble. 

The  property  of  hardening  under  water,  or  when 
excluded  from  air,  conferred  upon  a  paste  of  lime,  is 
effected  by  the  presence  of  foreign  substances — as 
silicum,  alumina,  iron,  etc.,  when  their  aggregate 
presence  amounts  to  one  tenth  of  the  whole. 

Limes  are  classed  :  1.  The  common  or  fat  limes. 
2.  The  poor  or  meagre.  3.  The  hydraulic.  4  The 
hydraulic  cements.  5.  The  natural  pozzuolanas,  in- 
cluding pozzuolana.,  properly  so  called,  trass  or  terras, 
the  arenes,  ochreous  earths,  basaltic  sands,  and  a 
variety  of  similar  substances. 

Rich  limes  are  fully  dissolved  in  water  frequently 
renewed,  and  they  remain  a  long  time  without  hard- 


COMMERCIAL    ARITHMETIC.  139 

«ning;  they  also  increase  greatly  in  volume — from 
t-»vo  to  three  and  a  half  times  their  original  bulks — 
and  will  not  harden  without  the  action  of  the  air. 
They  are  rendered  hydraulic  by  the  admixture  of 
pozzuolana  or  trass. 

Rich,  fat,  or  common  limes  usually  contain  less 
than  10  per  cent,  of  impurities.  j 

Hydraulic  limestones  are  those  which  contain  iron 
and  clay,  so  as  to  enable  them  to  produce  cements 
which  become  solid  when  under  water. 

The  pastes  of  fat  limes  shrink,  in  hardening,  to 
Buch  a  degree  that  they  cannot  be  used  as  mortar 
without  a  large  dose  of  sand. 

Poor  limes  have  all  the  defects  of  rich  limes,  and 
increase  but  slightly  in  bulk. 

The  poorer  limes  are  invariably  the  basis  of  the 
most  rapidly  setting  and  most  durable  cements  and 
mortars,  and  they  are  also  the  only  limes  which  have 
the  property,  when  in  combination  with  silica,  etc., 
of  indurating  under  water,  and  are,  therefore,  appli- 
cable for  the  admixture  of  hydraulic  cements  or  mor- 
tars. Alike  to  rich  limes,  they  will  not  harden  if 
in  a  state  of  paste  under  water-  or  in  wet  soil,  or  if 
excluded  from  contact  with  the  atmosphere  or  car- 
bonic acid  gas.  They  should  be  employed  for  mor- 
tar only  when  it  is  impracticable  to  procure  com- 
mon or  hydraulic  lime  or  cement,  in  which  case  it 
Is  recommended  to  reduce  them  to  powder  by  grind- 
ing. 

Lime  absorbs,  in  slaking,  a  mean  of  two  and  a  half 
times  its  volume,  and  two  and  a  quarter  times  its 
weight  of  water. 

Hydraulic  limes  are  those  which  readily  harden 
under  water.  Tlie  most  valuable  or  eminently  hy- 
draulic set  from  tlio  second  to  tlie  fourth  diiy  after 
immersion;  at  the  end  of  a  month  they  become  hard 


140  THE    MERCHANTS    AND    MECHANICS' 

and  insoluble,  and  at  the  end  of  six  months  they  are 
capable  of  being  worked,  like  the  hard,  natural  lime- 
stones. They  absorb  less  water  than  the  pure  limes, 
and  only  increase  in  bulk  from  one  and  three  quar- 
ters to  two  and  a  half  times  their  original  volume. 

The  inferior  grades,  or  moderately  hydraulic,  re- 
quire a  longer  period,  say  from  fifteen  to  twenty 
days'  immersion,  and  continue  to  harden  for  a  period 
of  six  months. 

The  resistance  of  nydraulic  limes  increases  if  sand 
is  mixed  in  the  propertion  of  50  to  180  per  cent,  of 
the  part  in  volume;  from  thence  it  decreases. 

Slaked  lime  is  a  hydrate  of  lime. 

M.  Vicat  declares  that  lime  is  rendered  hydraulic 
by  the  admixture  with  it  of  from  33  to  40  per  cent, 
of  clay  and  silica,  and  that  a  lime  is  obtained  which 
does  not  slake,  and  which  quickly  sets  under  water. 

Artificial  hydraulic  limes  do  not  attain,  even  under 
favorable  circumstances,  the  same  degree  of  hard- 
ness and  power  of  resistance  to  compression  as  the 
natural  limes  of  the  same  class. 

The  close  grained  and  densest  limestones  furnish 
the  best  limes. 

Hydraulic  limes  lose  or  depreciate  in  value  bv  ex- 
posure to  the  air. 

Arenes  is  a  species  of  ochreous  sand.  It  is  found 
in  France.  On  account  of  the  large  proportion  of 
clay  it  contains,  sometimes  as  great  as  seven  tenths, 
it  can  be  made  into  a  paste  with  water  without  any 
addition  of  lime;  hence,  it  is  sometimes  used  in  tliat 
state  for  walls  constructed  en  pise,  as  well  as  for 
mortar.  Mixed  Avith  rich  lime  it  gives  excellent  mor- 
tar, which  attains  great  hardness  under  water,  find 
possesses  great  hydraulic  energy. 

Pozzuolana  is  of  volcanic  origin.  It  comprises 
trass  or  terras,  the  arenes,   some  of  the  ochreous 


COMMERCIAL    ARITHMETIO.  141 

earths,  and  the  sand  of  certain  gray  wackes,  granites, 
schists  and  basalts;  their  principal  elements  are  sil- 
ica and  alumina,  the  former  preponderating.  None 
contain  more  than  10  per  cent,  of  lime. 

When  finely  pulverized,  without  previous  calcina- 
tion, and  combined  with  the  paste  of  fat  lime  in  pro- 
portions suitable  to  supply  its  deficiency  in  that 
element,  it  possesses  hydraulic  energy  to  a  valuable 
degree.  It  is  used  in  combination  with  rich  lime, 
and  may  be  made  by  slightly  calcining  clay  and 
driving  off  the  water  of  combination  at  a  tempera- 
ture of  1200°. 

Brick  or  tile  dust  combined  with  rich  lime  pos- 
sesses hydraulic  energy. 

Trass  or  terras  is  a  blue-black  trap,  and  is  also  of 
volcanic  origin.  It  requires  to  be  pulverized  and 
combined  with  rich  lime  to  render  it  fit  for  use,  and 
to  develop  any  of  its  hydraulic  properties. 

General  Gilmore*  designates  the  varieties  of  hy- 
draulic limes  as  follows  :  "  If,  after  being  slaked, 
they  harden  under  water  in  periods  varying  from  fif- 
teen to  twenty  days  after  immersion,  the}^  are  slightly 
hydraulic;  if  from  six  to  eight  days,  hydraulic;  and 
if  from  one  to  four  days,  eminently  hydraulic." 

Pulverized  silica  burned  with  rich  lime  produces 
hydraulic  lime  of  excellent  quality.  Hydraulic  limes 
are  injured  by  air  slaking  in  a  ratio  varying  directly 
with  their  hydraulicity,  and  they  deteriorate  by  age. 

For  foundations  in  a  damp  soil  or  exposure  hy- 
draulic limes  must  be  exclusively  employed. 

Cements. 
Hydraulic  cements  contain  a  larger  proportion  ot 
silica,  alumina,  magnesia,  etc.,  than  any  of  the  pre- 

*Sce  hl»  TreaUscB  on  LimcH,  Hydraulic  Cfraeuts  ami  Mortan,  and 
Papers  on  Practical  Engineering,  Engineer  Department  U.  8.  ▲. 


142  THE    MERCHANTS    AND    MECHANICS' 

ceding  varieties  of  lime.  They  do  not  slake  after 
calcination,  and  are  superior  to  the  very  best  of  hy- 
draulic limes,  as  some  of  them  set  under  water  at  a 
moderate  temperature  (65°)  in  from  three  to  four 
minutes;  others  require  as  many  hours.  They  do 
not  shrink  in  hardening,  and  make  an  excellent  mor- 
tar without  any  admixture  of  ^and. 

Roman  cement  is  made  from  a  lime  of  a  peculiar 
character,  found  in  England  and  France,  derived 
from  argillo  calcarious  kidney  shaped  stones,  termed 
"  Septaria." 

Rosendale  cement  is  from  Rosendale,  New  York. 

Portland  cement  is  made  in  England  and  France. 
It  requires  less  water  than  the  Roman  cement,  sets 
slowly,  and  can  be  remixed  with  additional  water 
after  an  interval  of  twelve  or  even  twenty-four  hours 
from  its  first  mixture. 

The  property  of  setting  slowly  may  be  an  obstacle 
to  the  use  of  some  designations  of  this  cement — as  the 
Boulogne,  when  required  for  localities  having  to  con- 
tend against  immediate  causes  of  destruction — as  in 
sea  constructions  having  to  be  executed  under  water 
and  between  tides.  On  tho  other  hand,  a  quick  set- 
ting cement  is  always  difficult  of  use;  it  requires 
special  workmen  and  an  active  supervision. 

Artificial  cement  is  made  by  combining  slaked 
lime  with  unburned  clay  in  suitable  proportions. 

Artificial  pozzuolana  is  made  by  subjecting  clay 
to  a  slight  calcination. 

Salt  water  has  a  tendency  to  decompose  cements 
of  all  kinds. 

Mortars. 

Lime  or  cement  paste  is  the  cementing  substance 
in  mortar,  and  its  proportion  should  be  determined 
by   the   rule   that  the  volume  of  the  cementing  sub- 


COMMERCIAL    ARITHMETIC.  143 

ttance  should  be  somewhat  in  excess  of  the  volume  of 
voids  or  spaces  in  the  sand  or  coarse  material  to  be 
united,  the  excess  being  added  to  meet  imperfect 
manipulation  of  the  mass. 

Hydraulic  mortar,  if  repulverized  and  formed  into 
a  paste  after  having  once  set,  immediately  loses  a 
great  portion  of  its  hydraulicity  and  descends  to  the 
level  of  the  moderate  hydraulic  limes. 

All  mortars  are  much  improved  by  being  worked 
or  manipulated;  and  as  rich  limes  gain  somewhat 
by  exposure  to  the  air,  it  is  advisable  to  work  mor- 
tar in  large  quantities,  and  then  render  it  fit  for  use 
by  a  second  manipulation. 

For  an  analysis  of  limestones,  etc.,  etc.,  see  Gen. 
,  Gilmore's  Treatise,  pp.  22-125. 

White  lime  will  take  a  larger  proportion  of  sand 
than  brown  lime. 

The  use  of  salt  water  in  the  composition  of  mop 
tar  injures  the  adhesion  of  it. 

When  a  small  quantity  of  water  is  mixed  with 
slaked  lime  a  stiff  paste  is  made,  which,  upon  be- 
coming dry  or  hard,  has  but  very  little  tenacity; 
but,  by  being  mixed  with  sand  or  like  substances,  it 
acquires  the  properties  of  a  cement  or  mortar. 

The  proportion  of  sand  that  can  be  incorporated 
with  mortar  depends  partly  upon  the  degree  of  fine- 
ness of  the  sand  itself,  and  partly  upon  the  charac- 
ter of  the  lime.  For  the  rich  limes  the  resistance  is 
increased  if  the  sand  is  in  proportions  varying  from 
50  to  240  per  cent,  of  the  paste  in  volume;  beyond 
this  proportion  the  resistance  decreases. 

Stone  Mortar. — 8  parts  cement,  3  parts  lime  and 
31  parts  of  sand. 

Brick  Mortar. — 8  parts  cement,  3  parts  lime  and 
17  parts  of  Band. 


144  THE    MERCHANTS    AND    MECHANICS' 

Brown  Mortar. — Lime,  1  part;  sand,  2  parts;  and 
a  small  quantity  of  hair. 

Lime  and  sand,  and  cement  and  sand,  lessen  about    ' 
one  third  in  volume  when  mixed  together. 

Calcareous  Mortar,  being  composed  of  one  or  more 
of  the  varieties  of  lime  or  cement,  natural  or  artifi- 
cial, mixed  with  sand,  will  vary  in  its  propertiesl 
with  the  quahty  of  the  lime  or  cement  used,  the  na- 
ture and  quality  of  the  sand,  and  the  method  of  man- 
ipulation. 

Mortar. — Lime,  1;  clean  sharp  sand,  2 J.  An  ex- 
cess of  water  in  slaking  the  lime  swells  the  mortar, 
which  remains  light  and  porous  or  shrinks  in  drying; 
an  excess  of  sand  destroys  the  cohesive  properties 
of  the  mass.  It  is  indispensable  that  the  sand 
should  be  sharp  and  clean. 

Turkish  Plaster,  or  Hydraulic  Cement. — 100  lbs. 
fresh  lime  reduced  to  powder,  10  quarts  linseed  oil 
and  one  to  two  ounces  cotton.  Manipulate  the  lime, 
gradually  mixing  the  oil  and  cotton,  in  a  wooden 
vessel,  until  the  mixture  becomes  of  the  consistency 
of  bread  dough.  Dry,  and,  when  required  for  use, 
mix  with  linseed  oil  to  the  consistency  of  paste,  and 
then  lay  on  in  coats.  Water  pipes  of  clay  or  metal, 
joined  or  coated  with  it,  resist  the  effect  of  humidity 
for  very  long  periods. 

Exterior  Plaster,  or  Stucco. — One  volume  of  cement 
powder  to  two  volumes  of  dry  sand. 

In  India,  to  the  water  for  mixing  the  plaster  is 
added  1  lb.  of  sugar  or  molasses  to  8  imperial  gal- 
lons of  water  for  the  first  coat;  and  for  the  second, 
or  finishing,  1  lb.  sugar  to  2  gallons  water. 

Powdered  slaked  lime  and  smith's  forge  scales, 
mixed  with  blood  in  suitable  proportions,  make  a 
moderate  Hydraulic  mortar. 


fir 


